L(s) = 1 | + (0.928 − 0.371i)2-s + (0.723 − 0.690i)4-s + (0.995 + 0.0950i)5-s + (0.888 + 0.458i)7-s + (0.415 − 0.909i)8-s + (0.959 − 0.281i)10-s + (−0.723 + 0.690i)13-s + (0.995 + 0.0950i)14-s + (0.0475 − 0.998i)16-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (0.786 − 0.618i)20-s + (0.888 − 0.458i)23-s + (0.981 + 0.189i)25-s + (−0.415 + 0.909i)26-s + ⋯ |
L(s) = 1 | + (0.928 − 0.371i)2-s + (0.723 − 0.690i)4-s + (0.995 + 0.0950i)5-s + (0.888 + 0.458i)7-s + (0.415 − 0.909i)8-s + (0.959 − 0.281i)10-s + (−0.723 + 0.690i)13-s + (0.995 + 0.0950i)14-s + (0.0475 − 0.998i)16-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (0.786 − 0.618i)20-s + (0.888 − 0.458i)23-s + (0.981 + 0.189i)25-s + (−0.415 + 0.909i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.591670952 - 0.6226303460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.591670952 - 0.6226303460i\) |
\(L(1)\) |
\(\approx\) |
\(2.272628701 - 0.3671653061i\) |
\(L(1)\) |
\(\approx\) |
\(2.272628701 - 0.3671653061i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.928 - 0.371i)T \) |
| 5 | \( 1 + (0.995 + 0.0950i)T \) |
| 7 | \( 1 + (0.888 + 0.458i)T \) |
| 13 | \( 1 + (-0.723 + 0.690i)T \) |
| 17 | \( 1 + (-0.654 + 0.755i)T \) |
| 19 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.888 - 0.458i)T \) |
| 29 | \( 1 + (-0.327 + 0.945i)T \) |
| 31 | \( 1 + (0.235 - 0.971i)T \) |
| 37 | \( 1 + (-0.959 + 0.281i)T \) |
| 41 | \( 1 + (-0.786 - 0.618i)T \) |
| 43 | \( 1 + (0.995 - 0.0950i)T \) |
| 47 | \( 1 + (0.786 - 0.618i)T \) |
| 53 | \( 1 + (-0.841 + 0.540i)T \) |
| 59 | \( 1 + (-0.928 - 0.371i)T \) |
| 61 | \( 1 + (0.786 - 0.618i)T \) |
| 67 | \( 1 + (-0.786 - 0.618i)T \) |
| 71 | \( 1 + (0.654 + 0.755i)T \) |
| 73 | \( 1 + (-0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.580 + 0.814i)T \) |
| 83 | \( 1 + (-0.888 - 0.458i)T \) |
| 89 | \( 1 + (0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.995 + 0.0950i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.57515830667198230012466842650, −20.744873325360181740919129571164, −20.37599548448439179341920755381, −19.35974077959392286006274236907, −17.85983042798647960615034649027, −17.57741865752711040327298268311, −16.9004862955440328056780941477, −15.84845949002168131080296247607, −15.108698027428038044017166429130, −14.277293543497046816187026963029, −13.678652686197715556747674077944, −13.10291975178163310237038114598, −12.12606153177879618302661278078, −11.25148961995366384717446511981, −10.5469435652607359782224025072, −9.4679122624306564603823111641, −8.50889683561139773017422034672, −7.414539981413725311063384191161, −6.93787447594104848265197812334, −5.74619059607182220621619572293, −5.05184693137376997504049340700, −4.519149992864697805919026025799, −3.10748477079480939546362555790, −2.37505319797143942886047833985, −1.25772860280046002736834157633,
1.47164908506782423048558164744, 2.0319748590390468120458713106, 2.93701424477510432914300888525, 4.17456027702083879583961755625, 5.04134160414489181304798284727, 5.64702629020276192985471710442, 6.56385164434020220587464009938, 7.398158485258760645983870196355, 8.71693790590154835242845071762, 9.53041335065803826197174746296, 10.483158415676447116622121895584, 11.104151358007736100717268012812, 12.06752359032699332590206595804, 12.67901570604167385913290781129, 13.65371989371748875861610977102, 14.28740407914535677275880007858, 14.83766533077966915976806143894, 15.65645529164021847218343201315, 16.84714265982902296540776210800, 17.37756604642765436652477855280, 18.56706917679784563990815833354, 18.95353606478450293430772451415, 20.24023781796691013004171024303, 20.74296277636945034804522491140, 21.5094377890479285744015253500