| L(s) = 1 | + (1.79 − 1.79i)5-s + (1.02 + 3.83i)7-s + (−1.05 + 3.95i)11-s + (1.54 + 3.25i)13-s + (−1.68 + 2.91i)17-s + (−7.63 + 2.04i)19-s + (−1.54 − 2.67i)23-s − 1.41i·25-s + (7.02 − 4.05i)29-s + (0.618 + 0.618i)31-s + (8.70 + 5.02i)35-s + (−8.92 − 2.39i)37-s + (6.25 + 1.67i)41-s + (8.40 + 4.85i)43-s + (4.37 + 4.37i)47-s + ⋯ |
| L(s) = 1 | + (0.801 − 0.801i)5-s + (0.388 + 1.44i)7-s + (−0.319 + 1.19i)11-s + (0.428 + 0.903i)13-s + (−0.408 + 0.707i)17-s + (−1.75 + 0.469i)19-s + (−0.322 − 0.558i)23-s − 0.283i·25-s + (1.30 − 0.753i)29-s + (0.111 + 0.111i)31-s + (1.47 + 0.849i)35-s + (−1.46 − 0.392i)37-s + (0.977 + 0.261i)41-s + (1.28 + 0.740i)43-s + (0.637 + 0.637i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39421 + 0.928563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39421 + 0.928563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-1.54 - 3.25i)T \) |
| good | 5 | \( 1 + (-1.79 + 1.79i)T - 5iT^{2} \) |
| 7 | \( 1 + (-1.02 - 3.83i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.05 - 3.95i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.63 - 2.04i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.54 + 2.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.02 + 4.05i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.618 - 0.618i)T + 31iT^{2} \) |
| 37 | \( 1 + (8.92 + 2.39i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.25 - 1.67i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.40 - 4.85i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.37 - 4.37i)T + 47iT^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 + (4.03 - 1.07i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.17 + 11.8i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.166 - 0.622i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.788 - 0.788i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + (-0.0917 + 0.0917i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.46 + 12.9i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-6.54 + 1.75i)T + (84.0 - 48.5i)T^{2} \) |
| 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
−10.12734306109965808860143860770, −9.160216045986610850894844908401, −8.751016493209495025434203745649, −7.963519887687340279782867399256, −6.48129625830260231811587742754, −5.98190507810529795061824111786, −4.90028746799820373654680544722, −4.25740507705922591602906053996, −2.27927180634260617762385168223, −1.84942500545357187732907356296,
0.78280293658607724609184175670, 2.41787166938535934820320777784, 3.44837998344225563929652024316, 4.52597896782864703961760364029, 5.70271502511303480533420665920, 6.50632165148031500324153958638, 7.26932192364173959074260416084, 8.217785457135138524801706597320, 9.024275151044624378964900883812, 10.32299588620774394955190587074
