L(s) = 1 | + (0.987 − 0.156i)2-s + (0.987 − 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (0.649 + 0.760i)7-s + (0.891 − 0.453i)8-s + (0.951 − 0.309i)9-s + (−0.233 + 0.972i)11-s + (0.891 − 0.453i)12-s + (0.649 − 0.760i)13-s + (0.760 + 0.649i)14-s + (0.809 − 0.587i)16-s + (−0.923 − 0.382i)17-s + (0.891 − 0.453i)18-s + (0.0784 + 0.996i)19-s + ⋯ |
L(s) = 1 | + (0.987 − 0.156i)2-s + (0.987 − 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (0.649 + 0.760i)7-s + (0.891 − 0.453i)8-s + (0.951 − 0.309i)9-s + (−0.233 + 0.972i)11-s + (0.891 − 0.453i)12-s + (0.649 − 0.760i)13-s + (0.760 + 0.649i)14-s + (0.809 − 0.587i)16-s + (−0.923 − 0.382i)17-s + (0.891 − 0.453i)18-s + (0.0784 + 0.996i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(6.574541125 + 0.2864213368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.574541125 + 0.2864213368i\) |
\(L(1)\) |
\(\approx\) |
\(3.083476677 - 0.1087396417i\) |
\(L(1)\) |
\(\approx\) |
\(3.083476677 - 0.1087396417i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 3 | \( 1 + (0.987 - 0.156i)T \) |
| 7 | \( 1 + (0.649 + 0.760i)T \) |
| 11 | \( 1 + (-0.233 + 0.972i)T \) |
| 13 | \( 1 + (0.649 - 0.760i)T \) |
| 17 | \( 1 + (-0.923 - 0.382i)T \) |
| 19 | \( 1 + (0.0784 + 0.996i)T \) |
| 23 | \( 1 + (0.923 - 0.382i)T \) |
| 29 | \( 1 + (-0.707 + 0.707i)T \) |
| 31 | \( 1 + (-0.382 + 0.923i)T \) |
| 37 | \( 1 + (-0.923 + 0.382i)T \) |
| 41 | \( 1 + (-0.156 - 0.987i)T \) |
| 43 | \( 1 + (0.760 + 0.649i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.707 + 0.707i)T \) |
| 59 | \( 1 + (-0.707 - 0.707i)T \) |
| 61 | \( 1 + (0.707 - 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 + (0.382 - 0.923i)T \) |
| 73 | \( 1 + (0.649 + 0.760i)T \) |
| 79 | \( 1 + (0.891 - 0.453i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.852 + 0.522i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.49076159976217341352219487630, −16.87727986369407820340116465990, −16.18561872047032818153457053170, −15.496235059213601717012739602514, −15.06905465551824250906618494123, −14.2824544212872187301798170796, −13.71794119010907413551232963931, −13.35184712787191866250107843969, −12.89105976750477928178290676981, −11.64040930308596938723235074692, −11.07301499535738956950165570307, −10.782767486899509414635819890171, −9.69888374628011734912845287208, −8.75727936363730997461427960963, −8.39317460614793944257764959186, −7.42178550476048577229025285505, −7.06468855626817665410589380874, −6.212738449963319340127367730281, −5.307156814345356698336493845, −4.55271622193751619922840667186, −3.93500579374489302745884209640, −3.47975244784530074543487252671, −2.50334830054965213612622826919, −1.913837947967049030051153876230, −0.99246349080236512852057228389,
1.261463195496491988617900795053, 1.863121517457278483217104037878, 2.51347078585098415004387892735, 3.234288628993781645950938969264, 3.90953785841226669839764027796, 4.82563910024265672047257875438, 5.21916216637692969198696552768, 6.19835357843688448500464275802, 6.96329567872005873245602580333, 7.59601815019013640492877496535, 8.26769491002039234539063568000, 9.026555280467690007693199088513, 9.70283189697038816880598914236, 10.774164210639266420195029179972, 10.94484973360993801924291880401, 12.26088177969135017365220592030, 12.474397733081159612383597553, 13.05719617439279465620780565961, 13.94858929443078287968419511918, 14.36351691677117587910249645777, 15.01568507441939454789847134586, 15.64893014561241804349799047937, 15.79919026710019845203628681177, 17.04919593897782240724441372270, 17.891209351879365490638705631094