| L(s) = 1 | + (0.723 + 0.690i)2-s + (0.436 + 3.03i)3-s + (0.0475 + 0.998i)4-s + (0.622 − 1.36i)5-s + (−1.77 + 2.49i)6-s + (0.959 − 3.95i)7-s + (−0.654 + 0.755i)8-s + (−6.13 + 1.80i)9-s + (1.39 − 0.556i)10-s + (−1.16 − 1.63i)11-s + (−3.00 + 0.580i)12-s + (−0.604 + 1.74i)13-s + (3.42 − 2.19i)14-s + (4.40 + 1.29i)15-s + (−0.995 + 0.0950i)16-s + (−0.174 + 3.67i)17-s + ⋯ |
| L(s) = 1 | + (0.511 + 0.487i)2-s + (0.251 + 1.75i)3-s + (0.0237 + 0.499i)4-s + (0.278 − 0.609i)5-s + (−0.725 + 1.01i)6-s + (0.362 − 1.49i)7-s + (−0.231 + 0.267i)8-s + (−2.04 + 0.600i)9-s + (0.439 − 0.176i)10-s + (−0.351 − 0.493i)11-s + (−0.868 + 0.167i)12-s + (−0.167 + 0.484i)13-s + (0.914 − 0.587i)14-s + (1.13 + 0.334i)15-s + (−0.248 + 0.0237i)16-s + (−0.0424 + 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03997 + 1.06603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03997 + 1.06603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.723 - 0.690i)T \) |
| 67 | \( 1 + (7.02 + 4.19i)T \) |
| good | 3 | \( 1 + (-0.436 - 3.03i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.622 + 1.36i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.959 + 3.95i)T + (-6.22 - 3.20i)T^{2} \) |
| 11 | \( 1 + (1.16 + 1.63i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.604 - 1.74i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (0.174 - 3.67i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (1.61 + 6.63i)T + (-16.8 + 8.70i)T^{2} \) |
| 23 | \( 1 + (-3.69 + 2.90i)T + (5.42 - 22.3i)T^{2} \) |
| 29 | \( 1 + (3.43 - 5.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.55 - 7.39i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (4.88 + 8.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.86 - 2.50i)T + (23.7 - 33.3i)T^{2} \) |
| 43 | \( 1 + (0.728 + 0.468i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-5.16 - 2.06i)T + (34.0 + 32.4i)T^{2} \) |
| 53 | \( 1 + (6.05 - 3.89i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.51 + 2.89i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-5.24 + 7.36i)T + (-19.9 - 57.6i)T^{2} \) |
| 71 | \( 1 + (-0.328 - 6.88i)T + (-70.6 + 6.74i)T^{2} \) |
| 73 | \( 1 + (-5.05 + 7.09i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-3.81 + 0.734i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (4.02 - 0.384i)T + (81.5 - 15.7i)T^{2} \) |
| 89 | \( 1 + (2.39 - 16.6i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.146 + 0.254i)T + (-48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.82192640388724846111613969054, −12.81007038279744157977243018887, −10.94916817628709477445424599633, −10.67296493587387808875705657706, −9.223706632985606919282675182414, −8.493431934711252830747532476386, −6.94416457581065060614440261965, −5.16503716420473723168905408136, −4.53929693366474441384247665640, −3.42415909845698024570323092650,
1.98609577100281238510945446292, 2.83047391154172397527688842723, 5.42502762261076441280184244533, 6.30565268456131941105370137196, 7.53550222549792722959982781782, 8.565732342493262201116819370184, 9.972056883739879765564949087115, 11.55609403311401190773978238889, 12.06317948386778844062614845094, 12.92884518262543415947558746618
