L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (0.719 + 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.374 + 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (0.719 + 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.374 + 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8899170097 + 0.3992185244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8899170097 + 0.3992185244i\) |
\(L(1)\) |
\(\approx\) |
\(1.028132369 - 0.2253504240i\) |
\(L(1)\) |
\(\approx\) |
\(1.028132369 - 0.2253504240i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.719 - 0.694i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.615 - 0.788i)T \) |
| 11 | \( 1 + (-0.374 + 0.927i)T \) |
| 13 | \( 1 + (0.719 + 0.694i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.615 + 0.788i)T \) |
| 29 | \( 1 + (-0.719 + 0.694i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.438 + 0.898i)T \) |
| 43 | \( 1 + (0.241 + 0.970i)T \) |
| 47 | \( 1 + (-0.559 + 0.829i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.241 - 0.970i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.978 + 0.207i)T \) |
| 79 | \( 1 + (0.882 + 0.469i)T \) |
| 83 | \( 1 + (-0.719 + 0.694i)T \) |
| 89 | \( 1 + (-0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.374 + 0.927i)T \) |
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
−22.36164574151783133308696498370, −21.33873157333470822241810012652, −20.55166496453716162130570825302, −19.8480813887440005349121789944, −18.67376889431095304703468947115, −18.08862168552560087650608543337, −16.84598840432363042604769003515, −16.11873368650031560830096452102, −15.70040202188666551838722787826, −15.05031747933452182526800000081, −13.77626937459053893454834966641, −13.1972283333692758024349181741, −12.38865445895723646827477718656, −11.7180212894936857773737706087, −10.81325370121231515623450939327, −9.28435480171237479518718472828, −8.52914664525142459313893513754, −7.96444457467034642416249669682, −6.87734738235240463060886715753, −5.78413823028679355951359144150, −5.38268885048487109861458743007, −4.08375253852981760285048482113, −3.42062601457508373962012679896, −2.38708393963371702792206957418, −0.35232912389864701337871904430,
1.32241155684069839067383180479, 2.53911965169096962363179483250, 3.538160741230960739158891547036, 4.12084973674639999588542756715, 5.05613178779522471086874799278, 6.48243632487496401426224936674, 6.89783013056691877012773269128, 7.98873553397001282761167177934, 9.459812801741055731835345386326, 10.00920284648683884038415185899, 11.18210914786829512759640776732, 11.35388347346062432029599791295, 12.57631074252634066997791011540, 13.281537068339945670251435773483, 14.01944930753333409413685887092, 14.91187374907867917767128874110, 15.70371111352100609415925871143, 16.268016238878397847914745619961, 17.76322801050038101834928127683, 18.41753318512343926463315393800, 19.33069618093565099433426800861, 20.011984580857716462162809082630, 20.423991138529849743437459849845, 21.57456537549753958494127967082, 22.42040676405312557812245753748