***** Quadruple precision version: precision forced to 128 bits main steps: ALL runComputation: steps: ALL ****** GETTING DIVISORS FROM FILE Plan generation 0.000000 min. 2.000000 sec. 214.708000 millisec. COMPUTE ODD LNGammaOdd computation 0.000000 min. 15.000000 sec. 219.028000 millisec. -temp-LGodd.qd 0.000000 min. 0.000000 sec. 88.947000 millisec. FFT(LGodd) 0.000000 min. 35.000000 sec. 985.567000 millisec. -temp-fftLGodd.qd 0.000000 min. 0.000000 sec. 89.319000 millisec. AK computation 0.000000 min. 4.000000 sec. 451.868000 millisec. -temp-ak.qd 0.000000 min. 0.000000 sec. 90.172000 millisec. FFT(AK) 0.000000 min. 36.000000 sec. 44.397000 millisec. -temp-fftAK.qd 0.000000 min. 0.000000 sec. 90.330000 millisec. Computing the sum and minmax over odd characters *** indexes for minimal and maximal odd characters L'/L (1,chi) chiminodd index = 5080907 chimaxodd index = 741403 Kummer_realcorrection = -7.5945873853385716406659622523e+07 Kummer_res = +7.5945874389515887564654390209e+07 Kummer (r) = +1.7093790418728811527042951110e+00 log r = +5.3613017115799476768604093902e-01 H_correction = -5.5557820330966760081289571809e+07 log h1 = +2.0388054058549127483364818400e+07 log Pi_odd = +5.3613017115799476768604093902e-01 *** indexes for minimal and maximal odd characters L (1,chi) chiminoddL index = 3248943 chimaxoddL index = 6763303 *** indexes for minimal and maximal odd characters L'(1,chi) chiminoddLprime index = 5080907 chimaxoddLprime index = 6763303 *** indexes for minimal and maximal odd characters |b_chi| chiminoddbchi index = 3507 chimaxoddbchi index = 3887129 *** First: oddresult = -8171246.179580045692567964821493826169 , (res = -1.207547569982381020035577293572 ) OddResult computation 0.000000 min. 6.000000 sec. 165.301000 millisec. COMPUTE EVEN loadTable(): initialize table INFO: precomputed table allocated 138 elements INFO: init() time (seconds) = 0.000007 S computation 0.000000 min. 22.000000 sec. 979.049000 millisec. -temp-S.qd 0.000000 min. 0.000000 sec. 89.352000 millisec. Sfft 0.000000 min. 35.000000 sec. 826.736000 millisec. -temp-fftS.qd 0.000000 min. 0.000000 sec. 92.322000 millisec. mkLGeven 0.000000 min. 5.000000 sec. 245.061000 millisec. -temp-LGeven.qd 0.000000 min. 0.000000 sec. 90.314000 millisec. LGevenfft 0.000000 min. 35.000000 sec. 747.220000 millisec. -temp-fftLGeven.qd 0.000000 min. 0.000000 sec. 87.351000 millisec. Computing the sum and minmax over even characters ... *** indexes for minimal and maximal even characters L'/L (1,chi) chimineven index = 4950582 chimaxeven index = 876438 *** indexes for minimal and maximal even characters L(1,chi) chiminevenL index = 3039832 chimaxevenL index = 18102 *** indexes for minimal and maximal even characters L'(1,chi) chiminevenLprime index = 4950582 chimaxevenLprime index = 6748708 *** indexes for minimal and maximal even characters |b_chi| chiminevenbchi index = 6748708 chimaxevenbchi index = 2451000 log Pi_even = -3.0505647960603287328033093699e+00 *** Second: evenresult = -8171224.023659376148637063331288398468 , (res = -16342448.047318752297274126662576796936 ) EvenResult computation 0.000000 min. 5.000000 sec. 802.234000 millisec. Inverse plan generation time 0.000000 min. 2.000000 sec. 223.139000 millisec. INVERSE LGodd FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 241.694000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 35.000000 sec. 974.441000 millisec. -temp-fftLGodd_INV.qd 0.000000 min. 0.000000 sec. 91.663000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 91.731000 millisec. INVERSE AK FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 242.075000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 35.000000 sec. 948.562000 millisec. -temp-fftAK_INV.qd 0.000000 min. 0.000000 sec. 88.971000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 89.046000 millisec. INVERSE S FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 228.186000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 35.000000 sec. 942.334000 millisec. -temp-fftS_INV.qd 0.000000 min. 0.000000 sec. 89.927000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 89.991000 millisec. INVERSE LGeven FFT_INV(): Initial copy and normalization time 0.000000 min. 0.000000 sec. 224.543000 millisec. FFT_INV(): fftwq inverse plan execute time 0.000000 min. 35.000000 sec. 999.390000 millisec. -temp-fftLGeven_INV.qd 0.000000 min. 0.000000 sec. 89.800000 millisec. FFT_INV(): Result copy 0.000000 min. 0.000000 sec. 89.859000 millisec. NORM OF S ... Measures for S NDIFF_2 = Norm2(S - fft(fft(S)) = +1.0368740838636507010643459056e-29 NBASE_2 = Norm2(S) = +1.2785104872709742970441093233e+04 Inverse precision for S in Norm2: NDIFF_2 / NBASE_2 = +8.1100162586612408881527084546e-34 NDIFF_1 = Norm1(S - fft(fft(S)) = +1.6371565923859282985645888505e-26 NBASE_1 = Norm1(S) = +1.3576480346355946063358650649e+07 Inverse precision for S in Norm1: NDIFF_1 / NBASE_1 = +1.2058770392764987839964596529e-33 NDIFF_0 = Norm00(S - fft(fft(S)) = +1.2359842804724238070062138465e-31 NBASE_0 = Norm00(S) = +2.4735552971252569739990199272e+02 Inverse precision for S in Norm0: NDIFF_0 / NBASE_0 = +4.9967926001447118444052787706e-34 NORM OF AK ... Measures for AK NDIFF_2 = Norm2(AK - fft(fft(AK)) = +9.3709765534312906799038075781e-31 NBASE_2 = Norm2(AK) = +1.0619799119255423060021187559e+03 Inverse precision for AK in Norm2: NDIFF_2 / NBASE_2 = +8.8240619697223707804594867827e-34 NDIFF_1 = Norm1(AK - fft(fft(AK)) = +1.5252682591677752050148351699e-27 NBASE_1 = Norm1(AK) = +1.6917022500000369450247686835e+06 Inverse precision for AK in Norm1: NDIFF_1 / NBASE_1 = +9.0161744430365443726047151282e-34 NDIFF_0 = Norm00(AK - fft(fft(AK)) = +2.0107298934091792405853897016e-33 NBASE_0 = Norm00(AK) = +9.9999970443980185053195663363e-01 Inverse precision for AK in Norm0: NDIFF_0 / NBASE_0 = +2.0107304877010806106932915459e-33 NORM OF LGODD ... Measures for LGodd NDIFF_2 = Norm2(LGodd - fft(fft(LGodd)) = +2.9311538228605304537484312090e-30 NBASE_2 = Norm2(LGodd) = +3.3236569412828985645293580809e+03 Inverse precision for LGodd in Norm2: NDIFF_2 / NBASE_2 = +8.8190624804048945980803159189e-34 NDIFF_1 = Norm1(LGodd - fft(fft(LGodd)) = +4.7388569995745259886561741627e-27 NBASE_1 = Norm1(LGodd) = +4.6589484812629683910284810899e+06 Inverse precision for LGodd in Norm1: NDIFF_1 / NBASE_1 = +1.0171516209361249778255682490e-33 NDIFF_0 = Norm00(LGodd - fft(fft(LGodd)) = +9.5289751058982243267654161985e-33 NBASE_0 = Norm00(LGodd) = +1.5727540314563708585510007002e+01 Inverse precision for LGodd in Norm0: NDIFF_0 / NBASE_0 = +6.0587828200156572413330012653e-34 NORM OF LGEVEN ... Measures for LGeven NDIFF_2 = Norm2(LGeven - fft(fft(LGeven)) = +1.6860643619204212291208308851e-30 NBASE_2 = Norm2(LGeven) = +2.0995646494657215163927369064e+03 Inverse precision for LGeven in Norm2: NDIFF_2 / NBASE_2 = +8.0305427239379165963273381149e-34 NDIFF_1 = Norm1(LGeven - fft(fft(LGeven)) = +2.6985515832109888332502429823e-27 NBASE_1 = Norm1(LGeven) = +2.3451897726721775642398590300e+06 Inverse precision for LGeven in Norm1: NDIFF_1 / NBASE_1 = +1.1506751456348799122368184818e-33 NDIFF_0 = Norm00(LGeven - fft(fft(LGeven)) = +9.4093647052450680125216789134e-33 NBASE_0 = Norm00(LGeven) = +1.4582810599316320667270359574e+01 Inverse precision for LGeven in Norm0: NDIFF_0 / NBASE_0 = +6.4523670805175258976038342922e-34 *** RESULTS: -------------- gamma_Kr constants for every intermediate field ---- gammaK_1(6766811) = 1.604045276207209271774295781554 gammaKplus_1(6766811) = 10.961044439670897364342780031279 ---- gammaK_5(6766811) = 14.508775480150269938531799241076 gammaKplus_5(6766811) = 14.010464182809112891100655388199 ---- gammaK_257(6766811) = 5.405179964627702811924273264620 gammaKplus_257(6766811) = -5.638529062214059434833093348850 ---- gammaK_1285(6766811) = 10.285906182601945196856424798000 gammaKplus_1285(6766811) = 6.942877258253508289418362158729 ---- gammaK_2633(6766811) = 9.539389310346796819856044417014 gammaKplus_2633(6766811) = 10.057097702222752244880520434899 ---- gammaK_13165(6766811) = 9.715243050561573491570681444207 gammaKplus_13165(6766811) = 8.146392716587937571335483591627 ---- gammaK_676681(6766811) = 2.820129244256749726459395307358 gammaKplus_676681(6766811) = 2.811885966383996042491899023552 ---- gammaK_3383405(6766811) = -0.896306122218629920784769461769 gammaKplus_3383405(6766811) = 0.577215664901532865549427242513 -------------- Other quantities -------------- [1] EK(6766811) = 1.604045276207209271774295781554 [2] EKplus(6766811) = 10.961044439670897364342780031279 [3] EK(6766811)diff = -9.356999163463688092568484248916 -------------- [4] min_even(6766811) = 0.000134627020627506619110078562 [5] min_odd(6766811) = 0.000316765756187305661719566886 [6] min_{chi neq chi_0}|L'/L(1,chi)|(6766811) = 0.000134627020627506619110078562 -------------- [7] max_even(6766811) = 2.756199328842647598783721974939 [8] max_odd(6766811) = 2.938078253126929987801206582066 [9] max_{chi neq chi_0}|L'/L(1,chi)|(6766811) = 2.938078253126929987801206582066 -------------- [10] Kummer_ratio(6766811) = 1.709379041872881152704295111017 [11] log(Kummer) = log r(6766811) = 0.536130171157994767686040939017 [12] logh_1(6766811) = 20388054.058549127483364818399824871380 -------------- [13] Pi(6766811,1.0Q) = 12.359618982580541559991409208716 [14] Laurent series of zeta_{Q(zeta_q)} at 1: c_{-1} = Pi(q,1)^(-1) = 0.080908643009900608129462180113 [15] c_{0} = c_{-1} * EK = 0.129781126624366512697918978198 -------------- [16] minL_even(6766811) = 0.262177518784878899654295099193 [17] minL_odd(6766811) = 0.250481879805769864570706466933 [18] minL_{chi neq chi_0}|L(1,chi)|(6766811) = 0.250481879805769864570706466933 -------------- [19] maxL_even(6766811) = 5.459526096967488462579220339598 [20] maxL_odd(6766811) = 5.593797993044597828021840058686 [21] maxL_{chi neq chi_0}|L(1,chi)|(6766811) = 5.593797993044597828021840058686 -------------- [22] minLprime_even(6766811) = 0.000151521253358665499124419201 [23] minLprime_odd(6766811) = 0.000256343704472125870975601711 [24] minLprime_{chi neq chi_0}|Lprime(1,chi)|(6766811) = 0.000151521253358665499124419201 -------------- [25] maxLprime_even(6766811) = 13.548348968878168224913698645224 [26] maxLprime_odd(6766811) = 15.239850452526253591762723711847 [27] maxLprime_{chi neq chi_0}|Lprime(1,chi)|(6766811) = 15.239850452526253591762723711847 -------------- [28] minbchi_even(6766811) = 10.831663753232337392592719568581 [29] minbchi_odd(6766811) = 10.640014760139021950546433773226 [30] minbchi_{chi neq chi_0}(6766811) = 10.640014760139021950546433773226 -------------- [31] maxbchi_even(6766811) = 15.727553156508781681521483256585 [32] maxbchi_odd(6766811) = 15.754164075696127842258097956385 [33] maxbchi_{chi neq chi_0}(6766811) = 15.754164075696127842258097956385 -------------- ******** the quadratic character is odd [34] L(1,chi_quad)(6766811) = 1.452859029009931016672102957818 [35] L(1,chi_quad)/log(6766811) = 0.092376747043205947693069931197 [36] L(1,chi_quad)/loglog(6766811) = 0.527274439891434328339759725181 [37] Lprime(1,chi_quad)(6766811) = -2.140819432860377981510032775365 [38] Lderlog(1,chi_quad)(6766811) = -1.473521787120162786334196704282 -------------- *** A priori (closed formulas) norms *** N1LGeven = +2.3451897726721791767663708526e+06 N00LGeven = +1.4582810599316320667270359574e+01 N00LGodd = +1.5727540314563708593466738315e+01 N1Seven = +1.3576480346355946499117883888e+07 N00Seven = +2.4735552971252569739990199273e+02 N1AKodd = +1.6917022500000369450247686835e+06 N00AKodd = +9.9999970443980185053195663363e-01 N2AKodd = +1.0619799119255423060021187559e+03 TOTAL TIME 6.000000 min. 14.000000 sec. 458.497000 millisec.