***** Quadruple precision version: precision forced to 128 bits main steps: ALL runComputation: steps: ALL ****** GETTING DIVISORS FROM FILE Plan generation 1.000000 min. 2.000000 sec. 768.342000 millisec. COMPUTE ODD LNGammaOdd computation 4.000000 min. 22.000000 sec. 304.974000 millisec. -temp-LGodd.qd 0.000000 min. 0.000000 sec. 969.787000 millisec. FFT(LGodd) 12.000000 min. 11.000000 sec. 639.670000 millisec. -temp-fftLGodd.qd 0.000000 min. 0.000000 sec. 999.317000 millisec. AK computation 1.000000 min. 16.000000 sec. 721.411000 millisec. -temp-ak.qd 0.000000 min. 0.000000 sec. 975.732000 millisec. FFT(AK) 12.000000 min. 11.000000 sec. 168.119000 millisec. -temp-fftAK.qd 0.000000 min. 0.000000 sec. 983.791000 millisec. Computing the sum and minmax over odd characters *** indexes for minimal and maximal odd characters L'/L (1,chi) chiminodd index = 81138049 chimaxodd index = 110767845 Kummer_realcorrection = -1.5607900662035876031669228979e+09 Kummer_res = +1.5607900656513742913350157855e+09 Kummer (r) = +5.7567425266296897380237451320e-01 log r = -5.5221331183190711243430682900e-01 H_correction = -1.1255946088043441263053426298e+09 log h1 = +4.3519545684703016502967315568e+08 log Pi_odd = -5.5221331183190711243430682900e-01 *** indexes for minimal and maximal odd characters L (1,chi) chiminoddL index = 49165037 chimaxoddL index = 723939 *** indexes for minimal and maximal odd characters L'(1,chi) chiminoddLprime index = 43855013 chimaxoddLprime index = 116103489 *** indexes for minimal and maximal odd characters |b_chi| chiminoddbchi index = 116103489 chimaxoddbchi index = 67817229 *** First: oddresult = -141074524.784239243020643366072423063233 , (res = -1.207546258543781212720544984966 ) OddResult computation 1.000000 min. 56.000000 sec. 782.448000 millisec. COMPUTE EVEN loadTable(): initialize table INFO: precomputed table allocated 138 elements INFO: init() time (seconds) = 0.000009 S computation 6.000000 min. 36.000000 sec. 765.265000 millisec. -temp-S.qd 0.000000 min. 0.000000 sec. 974.756000 millisec. Sfft 12.000000 min. 7.000000 sec. 722.466000 millisec. -temp-fftS.qd 0.000000 min. 1.000000 sec. 4.612000 millisec. mkLGeven 1.000000 min. 31.000000 sec. 290.909000 millisec. -temp-LGeven.qd 0.000000 min. 0.000000 sec. 974.642000 millisec. LGevenfft 12.000000 min. 7.000000 sec. 936.570000 millisec. -temp-fftLGeven.qd 0.000000 min. 0.000000 sec. 976.436000 millisec. Computing the sum and minmax over even characters ... *** indexes for minimal and maximal even characters L'/L (1,chi) chimineven index = 21736894 chimaxeven index = 12609430 *** indexes for minimal and maximal even characters L(1,chi) chiminevenL index = 48780924 chimaxevenL index = 1930166 *** indexes for minimal and maximal even characters L'(1,chi) chiminevenLprime index = 21736894 chimaxevenLprime index = 114897262 *** indexes for minimal and maximal even characters |b_chi| chiminevenbchi index = 114897262 chimaxevenbchi index = 49372760 log Pi_even = -3.1683848771520826320587933199e+00 *** Second: evenresult = -141074522.290182615883553946460293257661 , (res = -282149044.580365231767107892920586515323 ) EvenResult computation 1.000000 min. 51.000000 sec. 810.577000 millisec. Inverse plan generation time 1.000000 min. 2.000000 sec. 705.341000 millisec. INVERSE LGodd FFT_INV(): Initial copy and normalization time 0.000000 min. 3.000000 sec. 979.240000 millisec. FFT_INV(): fftwq inverse plan execute time 12.000000 min. 12.000000 sec. 502.382000 millisec. -temp-fftLGodd_INV.qd 0.000000 min. 1.000000 sec. 3.640000 millisec. FFT_INV(): Result copy 0.000000 min. 1.000000 sec. 3.659000 millisec. INVERSE AK FFT_INV(): Initial copy and normalization time 0.000000 min. 3.000000 sec. 510.293000 millisec. FFT_INV(): fftwq inverse plan execute time 12.000000 min. 10.000000 sec. 261.705000 millisec. -temp-fftAK_INV.qd 0.000000 min. 1.000000 sec. 24.436000 millisec. FFT_INV(): Result copy 0.000000 min. 1.000000 sec. 24.468000 millisec. INVERSE S FFT_INV(): Initial copy and normalization time 0.000000 min. 3.000000 sec. 685.862000 millisec. FFT_INV(): fftwq inverse plan execute time 12.000000 min. 10.000000 sec. 127.578000 millisec. -temp-fftS_INV.qd 0.000000 min. 2.000000 sec. 88.145000 millisec. FFT_INV(): Result copy 0.000000 min. 2.000000 sec. 88.170000 millisec. INVERSE LGeven FFT_INV(): Initial copy and normalization time 0.000000 min. 3.000000 sec. 413.031000 millisec. FFT_INV(): fftwq inverse plan execute time 12.000000 min. 9.000000 sec. 842.416000 millisec. -temp-fftLGeven_INV.qd 0.000000 min. 1.000000 sec. 30.705000 millisec. FFT_INV(): Result copy 0.000000 min. 1.000000 sec. 30.728000 millisec. NORM OF S ... Measures for S NDIFF_2 = Norm2(S - fft(fft(S)) = +4.0125719640862978046874254741e-29 NBASE_2 = Norm2(S) = +5.3130273307391632949709050535e+04 Inverse precision for S in Norm2: NDIFF_2 / NBASE_2 = +7.5523269772603588609179679039e-34 NDIFF_1 = Norm1(S - fft(fft(S)) = +2.7019824336575385710392003536e-25 NBASE_1 = Norm1(S) = +2.3439725771644793349245640892e+08 Inverse precision for S in Norm1: NDIFF_1 / NBASE_1 = +1.1527363673026185152344073837e-33 NDIFF_0 = Norm00(S - fft(fft(S)) = +9.8737257964762481036422791818e-32 NBASE_0 = Norm00(S) = +3.4507551993637661266832725094e+02 Inverse precision for S in Norm0: NDIFF_0 / NBASE_0 = +2.8613231672582044722146193131e-34 NORM OF AK ... Measures for AK NDIFF_2 = Norm2(AK - fft(fft(AK)) = +3.4501465774508373808204224507e-30 NBASE_2 = Norm2(AK) = +4.4126225384307085007791340442e+03 Inverse precision for AK in Norm2: NDIFF_2 / NBASE_2 = +7.8188119364450259355458129350e-34 NDIFF_1 = Norm1(AK - fft(fft(AK)) = +2.3357335457447061391237473900e-26 NBASE_1 = Norm1(AK) = +2.9206856750000002139908428525e+07 Inverse precision for AK in Norm1: NDIFF_1 / NBASE_1 = +7.9972095790304650567739904485e-34 NDIFF_0 = Norm00(AK - fft(fft(AK)) = +1.8937789819075465137772976335e-33 NBASE_0 = Norm00(AK) = +9.9999998288073257180041169955e-01 Inverse precision for AK in Norm0: NDIFF_0 / NBASE_0 = +1.8937790143276559099646589074e-33 NORM OF LGODD ... Measures for LGodd NDIFF_2 = Norm2(LGodd - fft(fft(LGodd)) = +1.0798737989485037462135879706e-29 NBASE_2 = Norm2(LGodd) = +1.3810181244640305726982588482e+04 Inverse precision for LGodd in Norm2: NDIFF_2 / NBASE_2 = +7.8194035242484588613697902288e-34 NDIFF_1 = Norm1(LGodd - fft(fft(LGodd)) = +7.3037167384473870506579541567e-26 NBASE_1 = Norm1(LGodd) = +8.0435810886628798634444763538e+07 Inverse precision for LGodd in Norm1: NDIFF_1 / NBASE_1 = +9.0801804046479952338233043055e-34 NDIFF_0 = Norm00(LGodd - fft(fft(LGodd)) = +7.8562834945945459980763895413e-33 NBASE_0 = Norm00(LGodd) = +1.8576208428235689308537523411e+01 Inverse precision for LGodd in Norm0: NDIFF_0 / NBASE_0 = +4.2292179940514973272290335118e-34 NORM OF LGEVEN ... Measures for LGeven NDIFF_2 = Norm2(LGeven - fft(fft(LGeven)) = +6.6002393395453473064037188184e-30 NBASE_2 = Norm2(LGeven) = +8.7240021601491179090767734104e+03 Inverse precision for LGeven in Norm2: NDIFF_2 / NBASE_2 = +7.5656094741642412888121510230e-34 NDIFF_1 = Norm1(LGeven - fft(fft(LGeven)) = +4.4538352190515480514345770878e-26 NBASE_1 = Norm1(LGeven) = +4.0489291877030750991090977538e+07 Inverse precision for LGeven in Norm1: NDIFF_1 / NBASE_1 = +1.1000032385298822380923135755e-33 NDIFF_0 = Norm00(LGeven - fft(fft(LGeven)) = +6.2171960219278326181448950513e-33 NBASE_0 = Norm00(LGeven) = +1.7431478552267798625091120694e+01 Inverse precision for LGeven in Norm0: NDIFF_0 / NBASE_0 = +3.5666486943639032373526025421e-34 *** RESULTS: -------------- gamma_Kr constants for every intermediate field ---- gammaK_1(116827429) = 23.268246056079740871743792619312 gammaKplus_1(116827429) = 11.962212808403742423388490653633 ---- gammaK_2(116827429) = 11.962212808403742423388490653633 gammaKplus_2(116827429) = 16.764015119424294997076468811185 ---- gammaK_3(116827429) = 19.912094665807961186048649450569 gammaKplus_3(116827429) = 16.306904469093115082521829718136 ---- gammaK_6(116827429) = 16.306904469093115082521829718136 gammaKplus_6(116827429) = 17.631853029213030598437501028977 ---- gammaK_19(116827429) = 19.752869572078872951159595359706 gammaKplus_19(116827429) = 16.195937641410558709455862984020 ---- gammaK_29(116827429) = 9.650303745160741695101072454105 gammaKplus_29(116827429) = 9.997354843068478165876720071845 ---- gammaK_38(116827429) = 16.195937641410558709455862984020 gammaKplus_38(116827429) = 16.929705049148499498274289418657 ---- gammaK_57(116827429) = 17.779701538228730808513509748175 gammaKplus_57(116827429) = 16.717441885730852027361105298579 ---- gammaK_58(116827429) = 9.997354843068478165876720071845 gammaKplus_58(116827429) = 13.993648648396786570481780019701 ---- gammaK_87(116827429) = 13.640519850053594045387277513775 gammaKplus_87(116827429) = 13.728316663702788287347611698150 ---- gammaK_114(116827429) = 16.717441885730852027361105298579 gammaKplus_114(116827429) = 17.187901939449891115822199313202 ---- gammaK_174(116827429) = 13.728316663702788287347611698150 gammaKplus_174(116827429) = 14.328957028178133473135212246688 ---- gammaK_551(116827429) = 11.439109887103691734540498336173 gammaKplus_551(116827429) = 5.211895487779436841848312737919 ---- gammaK_1102(116827429) = 5.211895487779436841848312737919 gammaKplus_1102(116827429) = 10.005220784765726798152600193467 ---- gammaK_1653(116827429) = 13.427922729615986220617909611496 gammaKplus_1653(116827429) = 11.529906102923725611790299873008 ---- gammaK_3306(116827429) = 11.529906102923725611790299873008 gammaKplus_3306(116827429) = 12.194579862621416427746521833952 ---- gammaK_17669(116827429) = 12.437892211269665356696255707603 gammaKplus_17669(116827429) = 9.378500330191120696398856585392 ---- gammaK_35338(116827429) = 9.378500330191120696398856585392 gammaKplus_35338(116827429) = -2.405029940596937171854288918472 ---- gammaK_53007(116827429) = 1.614485668799875653094714598857 gammaKplus_53007(116827429) = 2.693788934572033386410859072791 ---- gammaK_106014(116827429) = 2.693788934572033386410859072791 gammaKplus_106014(116827429) = 1.836770379779634278654735102094 ---- gammaK_335711(116827429) = 1.025345044956879030872274118610 gammaKplus_335711(116827429) = 3.503642336227602530158922082106 ---- gammaK_512401(116827429) = 8.695310843245041581480888271119 gammaKplus_512401(116827429) = 6.920765365909346436228947414401 ---- gammaK_671422(116827429) = 3.503642336227602530158922082106 gammaKplus_671422(116827429) = 4.331228004088282068081101515803 ---- gammaK_1007133(116827429) = 1.581391054434022648847694165295 gammaKplus_1007133(116827429) = 4.250906414867672838438140676139 ---- gammaK_1024802(116827429) = 6.920765365909346436228947414401 gammaKplus_1024802(116827429) = 5.773717545345548232790246526738 ---- gammaK_1537203(116827429) = 7.356444210456722368355382551112 gammaKplus_1537203(116827429) = 6.690069456750361692511277741169 ---- gammaK_2014266(116827429) = 4.250906414867672838438140676139 gammaKplus_2014266(116827429) = 4.144988883598296654062561561974 ---- gammaK_3074406(116827429) = 6.690069456750361692511277741169 gammaKplus_3074406(116827429) = 4.162907960067067164037783475849 ---- gammaK_9735619(116827429) = 3.242916403411113715507695222318 gammaKplus_9735619(116827429) = 0.494909861289919589820900702002 ---- gammaK_19471238(116827429) = 0.494909861289919589820900702002 gammaKplus_19471238(116827429) = 0.828967044191965933741221530501 ---- gammaK_29206857(116827429) = -0.821695736321458012004201901627 gammaKplus_29206857(116827429) = -0.475113637725766993114082654404 ---- gammaK_58413714(116827429) = -0.475113637725766993114082654404 gammaKplus_58413714(116827429) = 0.577215664901532865549427242513 -------------- Other quantities -------------- [1] EK(116827429) = 23.268246056079740871743792619312 [2] EKplus(116827429) = 11.962212808403742423388490653633 [3] EK(116827429)diff = 11.306033247675998448355301991529 -------------- [4] min_even(116827429) = 0.000023828131870840348110284808 [5] min_odd(116827429) = 0.000216858890455870026363456272 [6] min_{chi neq chi_0}|L'/L(1,chi)|(116827429) = 0.000023828131870840348110284808 -------------- [7] max_even(116827429) = 2.984078424291202912506276806440 [8] max_odd(116827429) = 3.098046611132524379182750338374 [9] max_{chi neq chi_0}|L'/L(1,chi)|(116827429) = 3.098046611132524379182750338374 -------------- [10] Kummer_ratio(116827429) = 0.575674252662968973802374513204 [11] log(Kummer) = log r(116827429) = -0.552213311831907112434306828998 [12] logh_1(116827429) = 435195456.847030165029673155679909570212 -------------- [13] Pi(116827429,1.0Q) = 41.289085398890225998627263442084 [14] Laurent series of zeta_{Q(zeta_q)} at 1: c_{-1} = Pi(q,1)^(-1) = 0.024219475688044137312023638434 [15] c_{0} = c_{-1} * EK = 0.563544719658652166468958697433 -------------- [16] minL_even(116827429) = 0.235677795864697927805644432975 [17] minL_odd(116827429) = 0.235379538241680188895623109242 [18] minL_{chi neq chi_0}|L(1,chi)|(116827429) = 0.235379538241680188895623109242 -------------- [19] maxL_even(116827429) = 6.014978129720257110618601775509 [20] maxL_odd(116827429) = 6.207381826762271798343843908001 [21] maxL_{chi neq chi_0}|L(1,chi)|(116827429) = 6.207381826762271798343843908001 -------------- [22] minLprime_even(116827429) = 0.000028353249793838048369046795 [23] minLprime_odd(116827429) = 0.000182911263632302336966754686 [24] minLprime_{chi neq chi_0}|Lprime(1,chi)|(116827429) = 0.000028353249793838048369046795 -------------- [25] maxLprime_even(116827429) = 17.026508514450034365951825986877 [26] maxLprime_odd(116827429) = 18.574852997051959725077595964191 [27] maxLprime_{chi neq chi_0}|Lprime(1,chi)|(116827429) = 18.574852997051959725077595964191 -------------- [28] minbchi_even(116827429) = 13.352683435698084530536397704438 [29] minbchi_odd(116827429) = 13.168772454283129661026574216089 [30] minbchi_{chi neq chi_0}(116827429) = 13.168772454283129661026574216089 -------------- [31] maxbchi_even(116827429) = 18.825870430325689132344425998360 [32] maxbchi_odd(116827429) = 18.743669769928841709651706064524 [33] maxbchi_{chi neq chi_0}(116827429) = 18.825870430325689132344425998360 -------------- ******** the quadratic character is even [34] L(1,chi_quad)(116827429) = 1.325578085631712710380173627273 [35] L(1,chi_quad)/log(116827429) = 0.071358915359267328144051916176 [36] L(1,chi_quad)/loglog(116827429) = 0.453672751452055007899080633745 [37] Lprime(1,chi_quad)(116827429) = -1.394944662430851411343115670298 [38] Lderlog(1,chi_quad)(116827429) = -1.052329302627299858663509896917 -------------- *** A priori (closed formulas) norms *** N1LGeven = +4.0489291877030786101340383948e+07 N00LGeven = +1.7431478552267798625091120694e+01 N00LGodd = +1.8576208428235689304043997728e+01 N1Seven = +2.3439725771644794101573006240e+08 N00Seven = +3.4507551993637661266832725087e+02 N1AKodd = +2.9206856750000002139908428525e+07 N00AKodd = +9.9999998288073257180041169955e-01 N2AKodd = +4.4126225384307085007791340442e+03 TOTAL TIME 123.000000 min. 30.000000 sec. 83.192000 millisec.