***** Quadruple precision version: precision forced to 128 bits main steps: ALL runComputation: steps: ALL ****** GETTING DIVISORS FROM FILE Plan generation 0.000000 min. 1.000000 sec. 510.390000 millisec. COMPUTE ODD LNGammaOdd computation 0.000000 min. 12.000000 sec. 572.086000 millisec. -temp-LGodd.qd 0.000000 min. 0.000000 sec. 76.547000 millisec. FFT(LGodd) 0.000000 min. 14.000000 sec. 957.425000 millisec. -temp-fftLGodd.qd 0.000000 min. 0.000000 sec. 75.644000 millisec. AK computation 0.000000 min. 3.000000 sec. 606.317000 millisec. -temp-ak.qd 0.000000 min. 0.000000 sec. 85.002000 millisec. FFT(AK) 0.000000 min. 15.000000 sec. 3.415000 millisec. -temp-fftAK.qd 0.000000 min. 0.000000 sec. 75.335000 millisec. Computing the sum and minmax over odd characters *** indexes for minimal and maximal odd characters L'/L (1,chi) chiminodd index = 3445221 chimaxodd index = 962203 Kummer_realcorrection = -6.1927371879172423641353287231e+07 Kummer_res = +6.1927371346395941269968957539e+07 Kummer (r) = +5.8697298499308276877721767961e-01 log r = -5.3277648237138432969212854820e-01 H_correction = -4.5354652259275566992963381346e+07 log h1 = +1.6572719087120374277005576193e+07 log Pi_odd = -5.3277648237138432969212854820e-01 *** indexes for minimal and maximal odd characters L (1,chi) chiminoddL index = 34647 chimaxoddL index = 370545 *** indexes for minimal and maximal odd characters L'(1,chi) chiminoddLprime index = 143301 chimaxoddLprime index = 5218623 *** indexes for minimal and maximal odd characters |b_chi| chiminoddbchi index = 5396045 chimaxoddbchi index = 1897499 *** First: oddresult = -6749170.330824295333507349408992204365 , (res = -1.207544508105640629851655838103 ) OddResult computation 0.000000 min. 6.000000 sec. 261.455000 millisec. COMPUTE EVEN loadTable(): initialize table INFO: precomputed table allocated 138 elements INFO: init() time (seconds) = 0.000015 S computation 0.000000 min. 19.000000 sec. 189.504000 millisec. -temp-S.qd 0.000000 min. 0.000000 sec. 74.644000 millisec. Sfft 0.000000 min. 12.000000 sec. 943.154000 millisec. -temp-fftS.qd 0.000000 min. 0.000000 sec. 75.883000 millisec. mkLGeven 0.000000 min. 4.000000 sec. 326.728000 millisec. -temp-LGeven.qd 0.000000 min. 0.000000 sec. 78.389000 millisec. LGevenfft 0.000000 min. 12.000000 sec. 867.640000 millisec. -temp-fftLGeven.qd 0.000000 min. 0.000000 sec. 83.757000 millisec. Computing the sum and minmax over even characters ... *** indexes for minimal and maximal even characters L'/L (1,chi) chimineven index = 5142046 chimaxeven index = 2888946 *** indexes for minimal and maximal even characters L(1,chi) chiminevenL index = 4937260 chimaxevenL index = 672030 *** indexes for minimal and maximal even characters L'(1,chi) chiminevenLprime index = 5142046 chimaxevenLprime index = 672030 *** indexes for minimal and maximal even characters |b_chi| chiminevenbchi index = 672030 chimaxevenbchi index = 5023840 log Pi_even = -3.0157162204930375352229305046e+00 *** Second: evenresult = -6749167.288417125044974035774817379991 , (res = -13498334.576834250089948071549634759981 ) EvenResult computation 0.000000 min. 6.000000 sec. 85.850000 millisec. Inverse plan generation time 0.000000 min. 1.000000 sec. 516.863000 millisec. INVERSE LGodd FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 203.980000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 14.000000 sec. 983.404000 millisec. -temp-fftLGodd_INV.qd 0.000000 min. 0.000000 sec. 76.943000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 77.007000 millisec. INVERSE AK FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 182.669000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 14.000000 sec. 981.200000 millisec. -temp-fftAK_INV.qd 0.000000 min. 0.000000 sec. 77.497000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 77.567000 millisec. INVERSE S FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 173.897000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 14.000000 sec. 773.960000 millisec. -temp-fftS_INV.qd 0.000000 min. 0.000000 sec. 75.665000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 75.727000 millisec. INVERSE LGeven FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 176.529000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 14.000000 sec. 896.667000 millisec. -temp-fftLGeven_INV.qd 0.000000 min. 0.000000 sec. 79.538000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 79.611000 millisec. NORM OF S ... Measures for S NDIFF_2 = Norm2(S - fft(fft(S)) = +5.6362850735047085664371319303e-30 NBASE_2 = Norm2(S) = +1.1619185404171265303582806816e+04 Inverse precision for S in Norm2: NDIFF_2 / NBASE_2 = +4.8508435638537044269351612521e-34 NDIFF_1 = Norm1(S - fft(fft(S)) = +8.1877928979959915478923797595e-27 NBASE_1 = Norm1(S) = +1.1213714057123965673865942409e+07 Inverse precision for S in Norm1: NDIFF_1 / NBASE_1 = +7.3015888012539117219693109382e-34 NDIFF_0 = Norm00(S - fft(fft(S)) = +2.6566296796790852989624748784e-32 NBASE_0 = Norm00(S) = +2.4137789712968132459695097336e+02 Inverse precision for S in Norm0: NDIFF_0 / NBASE_0 = +1.1006101682341691187025603395e-34 NORM OF AK ... Measures for AK NDIFF_2 = Norm2(AK - fft(fft(AK)) = +4.7539830081247808629543644347e-31 NBASE_2 = Norm2(AK) = +9.6515680936660562205613179465e+02 Inverse precision for AK in Norm2: NDIFF_2 / NBASE_2 = +4.9256068671832016194238052190e-34 NDIFF_1 = Norm1(AK - fft(fft(AK)) = +7.0332888805594645268375862867e-28 NBASE_1 = Norm1(AK) = +1.3972917500000447293685340343e+06 Inverse precision for AK in Norm1: NDIFF_1 / NBASE_1 = +5.0335149266853106238503226093e-34 NDIFF_0 = Norm00(AK - fft(fft(AK)) = +1.1271216024060766020382870456e-33 NBASE_0 = Norm00(AK) = +9.9999964216505172772553486932e-01 Inverse precision for AK in Norm0: NDIFF_0 / NBASE_0 = +1.1271220057297212188753227034e-33 NORM OF LGODD ... Measures for LGodd NDIFF_2 = Norm2(LGodd - fft(fft(LGodd)) = +1.4909825663220304343611992601e-30 NBASE_2 = Norm2(LGodd) = +3.0206278862160378448809163452e+03 Inverse precision for LGodd in Norm2: NDIFF_2 / NBASE_2 = +4.9360021243457265369166750047e-34 NDIFF_1 = Norm1(LGodd - fft(fft(LGodd)) = +2.2033316690462579791366217060e-27 NBASE_1 = Norm1(LGodd) = +3.8481406866676533233599662401e+06 Inverse precision for LGodd in Norm1: NDIFF_1 / NBASE_1 = +5.7257045634531133436319020375e-34 NDIFF_0 = Norm00(LGodd - fft(fft(LGodd)) = +3.6338373514024014629105850965e-33 NBASE_0 = Norm00(LGodd) = +1.5536340969214175521248410013e+01 Inverse precision for LGodd in Norm0: NDIFF_0 / NBASE_0 = +2.3389273951974807251076136348e-34 NORM OF LGEVEN ... Measures for LGeven NDIFF_2 = Norm2(LGeven - fft(fft(LGeven)) = +8.5334608763374481299359441141e-31 NBASE_2 = Norm2(LGeven) = +1.9081373518265195814138644658e+03 Inverse precision for LGeven in Norm2: NDIFF_2 / NBASE_2 = +4.4721418341132484509398558001e-34 NDIFF_1 = Norm1(LGeven - fft(fft(LGeven)) = +1.2477516825733432770908219340e-27 NBASE_1 = Norm1(LGeven) = +1.9370502522673448732592645543e+06 Inverse precision for LGeven in Norm1: NDIFF_1 / NBASE_1 = +6.4415039367865246881209833590e-34 NDIFF_0 = Norm00(LGeven - fft(fft(LGeven)) = +3.2550848537726561314817482549e-33 NBASE_0 = Norm00(LGeven) = +1.4391611289912765634823093687e+01 Inverse precision for LGeven in Norm0: NDIFF_0 / NBASE_0 = +2.2617931989687492045021016918e-34 *** RESULTS: -------------- gamma_Kr constants for every intermediate field ---- gammaK_1(5589169) = 19.553756872315036722735578327668 gammaKplus_1(5589169) = 10.379143488097112295895406002672 ---- gammaK_2(5589169) = 10.379143488097112295895406002672 gammaKplus_2(5589169) = 12.966390294653701359044839233348 ---- gammaK_3(5589169) = 18.915088032345875055803134019022 gammaKplus_3(5589169) = 14.806190602295518272872815673288 ---- gammaK_4(5589169) = 12.966390294653701359044839233348 gammaKplus_4(5589169) = 13.622642685792077234020147189442 ---- gammaK_6(5589169) = 14.806190602295518272872815673288 gammaKplus_6(5589169) = 14.981201640553746740782868141191 ---- gammaK_8(5589169) = 13.622642685792077234020147189442 gammaKplus_8(5589169) = 13.983815865654329770341399621366 ---- gammaK_12(5589169) = 14.981201640553746740782868141191 gammaKplus_12(5589169) = 15.641169090970652321123061060890 ---- gammaK_13(5589169) = 12.407028580431431411723072879519 gammaKplus_13(5589169) = 12.676523931561434994180003980322 ---- gammaK_24(5589169) = 15.641169090970652321123061060890 gammaKplus_24(5589169) = 15.071771773190258366639030592830 ---- gammaK_26(5589169) = 12.676523931561434994180003980322 gammaKplus_26(5589169) = -60.311793003170445224107633267329 ---- gammaK_39(5589169) = 14.064685015895239520567075459196 gammaKplus_39(5589169) = 13.551165080999559916311935268324 ---- gammaK_52(5589169) = -60.311793003170445224107633267329 gammaKplus_52(5589169) = -33.610449900092666467037023342203 ---- gammaK_53(5589169) = 9.625649628342950985476548648907 gammaKplus_53(5589169) = 11.454376977360917274712166624814 ---- gammaK_78(5589169) = 13.551165080999559916311935268324 gammaKplus_78(5589169) = -11.935961401628687137500600646057 ---- gammaK_104(5589169) = -33.610449900092666467037023342203 gammaKplus_104(5589169) = -9.873427239016986998176033702904 ---- gammaK_106(5589169) = 11.454376977360917274712166624814 gammaKplus_106(5589169) = 12.753403890658104883125186243844 ---- gammaK_156(5589169) = -11.935961401628687137500600646057 gammaKplus_156(5589169) = -3.583718488367871404535592799675 ---- gammaK_159(5589169) = 12.001622117146541051364817858409 gammaKplus_159(5589169) = 12.668040293249559815583091178562 ---- gammaK_169(5589169) = 12.489426317461294267177534258042 gammaKplus_169(5589169) = 12.148912179295737911267520440496 ---- gammaK_212(5589169) = 12.753403890658104883125186243844 gammaKplus_212(5589169) = 12.100783573336547918930500047768 ---- gammaK_312(5589169) = -3.583718488367871404535592799675 gammaKplus_312(5589169) = 4.040503199513202759407674049220 ---- gammaK_318(5589169) = 12.668040293249559815583091178562 gammaKplus_318(5589169) = 12.544308686034146324330650386951 ---- gammaK_338(5589169) = 12.148912179295737911267520440496 gammaKplus_338(5589169) = 5.801584752219400818191475000664 ---- gammaK_424(5589169) = 12.100783573336547918930500047768 gammaKplus_424(5589169) = 11.311686844918675154375243040794 ---- gammaK_507(5589169) = 6.474275617744327891857096258630 gammaKplus_507(5589169) = 9.218950751466138012690680940082 ---- gammaK_636(5589169) = 12.544308686034146324330650386951 gammaKplus_636(5589169) = 11.098010997385593776811836081589 ---- gammaK_676(5589169) = 5.801584752219400818191475000664 gammaKplus_676(5589169) = 7.738631785097268291810596556494 ---- gammaK_689(5589169) = 6.063797774630188899880386897617 gammaKplus_689(5589169) = 2.440408005946331267494305746383 ---- gammaK_1014(5589169) = 9.218950751466138012690680940082 gammaKplus_1014(5589169) = 7.270240663755114880846286222892 ---- gammaK_1272(5589169) = 11.098010997385593776811836081589 gammaKplus_1272(5589169) = 7.560325768542745449242396494626 ---- gammaK_1352(5589169) = 7.738631785097268291810596556494 gammaKplus_1352(5589169) = 9.613822328731248868724640946792 ---- gammaK_1378(5589169) = 2.440408005946331267494305746383 gammaKplus_1378(5589169) = 5.217382309994119525834381606463 ---- gammaK_2028(5589169) = 7.270240663755114880846286222892 gammaKplus_2028(5589169) = -9.956396226347328821171973387612 ---- gammaK_2067(5589169) = 6.984150163551679985540733229385 gammaKplus_2067(5589169) = 5.625014186866304830030235897739 ---- gammaK_2197(5589169) = 10.452281743199547724598592995404 gammaKplus_2197(5589169) = 9.517759445770697662180558265531 ---- gammaK_2756(5589169) = 5.217382309994119525834381606463 gammaKplus_2756(5589169) = 7.770640364366245028611744845579 ---- gammaK_4056(5589169) = -9.956396226347328821171973387612 gammaKplus_4056(5589169) = 0.117804829628554431529696815998 ---- gammaK_4134(5589169) = 5.625014186866304830030235897739 gammaKplus_4134(5589169) = 7.091560417220172504368668299600 ---- gammaK_4394(5589169) = 9.517759445770697662180558265531 gammaKplus_4394(5589169) = 7.852517412703273218234957757568 ---- gammaK_5512(5589169) = 7.770640364366245028611744845579 gammaKplus_5512(5589169) = 5.725892984869958067524952853767 ---- gammaK_6591(5589169) = 7.719331377044142031897684538869 gammaKplus_6591(5589169) = 6.879937513038022077333241914653 ---- gammaK_8268(5589169) = 7.091560417220172504368668299600 gammaKplus_8268(5589169) = 7.132009187930662335273324736996 ---- gammaK_8788(5589169) = 7.852517412703273218234957757568 gammaKplus_8788(5589169) = 7.900686988745776749074113417408 ---- gammaK_8957(5589169) = 6.368428321770062184825439498380 gammaKplus_8957(5589169) = 4.042316052380277606799002792210 ---- gammaK_13182(5589169) = 6.879937513038022077333241914653 gammaKplus_13182(5589169) = 6.671502789314392729172971387457 ---- gammaK_16536(5589169) = 7.132009187930662335273324736996 gammaKplus_16536(5589169) = 6.962158573333345006866296706251 ---- gammaK_17576(5589169) = 7.900686988745776749074113417408 gammaKplus_17576(5589169) = 6.534612259055769837593007719766 ---- gammaK_17914(5589169) = 4.042316052380277606799002792210 gammaKplus_17914(5589169) = 4.213434486047127162445548061177 ---- gammaK_26364(5589169) = 6.671502789314392729172971387457 gammaKplus_26364(5589169) = 5.235727677114118409271103728811 ---- gammaK_26871(5589169) = 6.894305289189309595514826796100 gammaKplus_26871(5589169) = 5.732436808032571527217903646059 ---- gammaK_35828(5589169) = 4.213434486047127162445548061177 gammaKplus_35828(5589169) = 2.557174206988120285021338862781 ---- gammaK_52728(5589169) = 5.235727677114118409271103728811 gammaKplus_52728(5589169) = 2.957775412854364803483213526143 ---- gammaK_53742(5589169) = 5.732436808032571527217903646059 gammaKplus_53742(5589169) = 4.882959225439015806752939650818 ---- gammaK_71656(5589169) = 2.557174206988120285021338862781 gammaKplus_71656(5589169) = -7.673065948325087318933171279350 ---- gammaK_107484(5589169) = 4.882959225439015806752939650818 gammaKplus_107484(5589169) = 2.879829278660985101045742516842 ---- gammaK_116441(5589169) = -2.802083317615772753227126583343 gammaKplus_116441(5589169) = -8.411492571866853290802738212992 ---- gammaK_214968(5589169) = 2.879829278660985101045742516842 gammaKplus_214968(5589169) = 0.743904802895238760238849982592 ---- gammaK_232882(5589169) = -8.411492571866853290802738212992 gammaKplus_232882(5589169) = -3.690608161836370400103875595545 ---- gammaK_349323(5589169) = 1.106669380818921236598852722358 gammaKplus_349323(5589169) = -2.808266254306758602721066895493 ---- gammaK_465764(5589169) = -3.690608161836370400103875595545 gammaKplus_465764(5589169) = -1.096480510303048079689821804368 ---- gammaK_698646(5589169) = -2.808266254306758602721066895493 gammaKplus_698646(5589169) = -1.531355971707979498429256727172 ---- gammaK_931528(5589169) = -1.096480510303048079689821804368 gammaKplus_931528(5589169) = -1.176127795019389124680373521363 ---- gammaK_1397292(5589169) = -1.531355971707979498429256727172 gammaKplus_1397292(5589169) = -0.931026560884777633713001760904 ---- gammaK_2794584(5589169) = -0.931026560884777633713001760904 gammaKplus_2794584(5589169) = 0.577215664901532865549427242513 -------------- Other quantities -------------- [1] EK(5589169) = 19.553756872315036722735578327668 [2] EKplus(5589169) = 10.379143488097112295895406002672 [3] EK(5589169)diff = 9.174613384217924426840172325804 -------------- [4] min_even(5589169) = 0.000800679898886384102062099823 [5] min_odd(5589169) = 0.001708932561202855535046698113 [6] min_{chi neq chi_0}|L'/L(1,chi)|(5589169) = 0.000800679898886384102062099823 -------------- [7] max_even(5589169) = 2.760086397665464877866702565600 [8] max_odd(5589169) = 2.765848939555914059892277284013 [9] max_{chi neq chi_0}|L'/L(1,chi)|(5589169) = 2.765848939555914059892277284013 -------------- [10] Kummer_ratio(5589169) = 0.586972984993082768777217679609 [11] log(Kummer) = log r(5589169) = -0.532776482371384329692128548203 [12] logh_1(5589169) = 16572719.087120374277005576192566966364 -------------- [13] Pi(5589169,1.0Q) = 34.760883000976089015182499774696 [14] Laurent series of zeta_{Q(zeta_q)} at 1: c_{-1} = Pi(q,1)^(-1) = 0.028767968868107288920098155838 [15] c_{0} = c_{-1} * EK = 0.562521868957297928986626438711 -------------- [16] minL_even(5589169) = 0.255252906568754884841275325546 [17] minL_odd(5589169) = 0.241217036209512687265251772472 [18] minL_{chi neq chi_0}|L(1,chi)|(5589169) = 0.241217036209512687265251772472 -------------- [19] maxL_even(5589169) = 5.432389901089017656592852332220 [20] maxL_odd(5589169) = 5.653739128457788295451892625583 [21] maxL_{chi neq chi_0}|L(1,chi)|(5589169) = 5.653739128457788295451892625583 -------------- [22] minLprime_even(5589169) = 0.000682399088781442149217152978 [23] minLprime_odd(5589169) = 0.001103965161758514375597436972 [24] minLprime_{chi neq chi_0}|Lprime(1,chi)|(5589169) = 0.000682399088781442149217152978 -------------- [25] maxLprime_even(5589169) = 13.530932245632711357779731983380 [26] maxLprime_odd(5589169) = 15.306544627183689411679043089648 [27] maxLprime_{chi neq chi_0}|Lprime(1,chi)|(5589169) = 15.306544627183689411679043089648 -------------- [28] minbchi_even(5589169) = 10.630810527049265690147426342374 [29] minbchi_odd(5589169) = 10.410055406959027573040094155104 [30] minbchi_{chi neq chi_0}(5589169) = 10.410055406959027573040094155104 -------------- [31] maxbchi_even(5589169) = 15.654836447951280509934480806488 [32] maxbchi_odd(5589169) = 15.605309310533399076669801456673 [33] maxbchi_{chi neq chi_0}(5589169) = 15.654836447951280509934480806488 -------------- ******** the quadratic character is even [34] L(1,chi_quad)(5589169) = 3.455083016479023204916151188922 [35] L(1,chi_quad)/log(5589169) = 0.222387174521451574842689222051 [36] L(1,chi_quad)/loglog(5589169) = 1.259516568039300924160671209732 [37] Lprime(1,chi_quad)(5589169) = -5.211102099050801676090126759645 [38] Lderlog(1,chi_quad)(5589169) = -1.508242225786310499262429003417 -------------- *** A priori (closed formulas) norms *** N1LGeven = +1.9370502522673463215158895912e+06 N00LGeven = +1.4391611289912765634823093687e+01 N00LGodd = +1.5536340969214175472897925714e+01 N1Seven = +1.1213714057123966033789190914e+07 N00Seven = +2.4137789712968132459695097335e+02 N1AKodd = +1.3972917500000447293685340343e+06 N00AKodd = +9.9999964216505172772553486932e-01 N2AKodd = +9.6515680936660562205613179465e+02 TOTAL TIME 3.000000 min. 9.000000 sec. 0.136000 millisec.