| L(s) = 1 | − 4·2-s + 13.8·3-s + 16·4-s + 25·5-s − 55.2·6-s + 213.·7-s − 64·8-s − 52.2·9-s − 100·10-s + 587.·11-s + 220.·12-s − 169·13-s − 852.·14-s + 345.·15-s + 256·16-s + 162.·17-s + 208.·18-s − 81.3·19-s + 400·20-s + 2.94e3·21-s − 2.34e3·22-s − 2.94e3·23-s − 883.·24-s + 625·25-s + 676·26-s − 4.07e3·27-s + 3.41e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.886·3-s + 0.5·4-s + 0.447·5-s − 0.626·6-s + 1.64·7-s − 0.353·8-s − 0.215·9-s − 0.316·10-s + 1.46·11-s + 0.443·12-s − 0.277·13-s − 1.16·14-s + 0.396·15-s + 0.250·16-s + 0.136·17-s + 0.152·18-s − 0.0517·19-s + 0.223·20-s + 1.45·21-s − 1.03·22-s − 1.16·23-s − 0.313·24-s + 0.200·25-s + 0.196·26-s − 1.07·27-s + 0.822·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.462464481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.462464481\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 - 13.8T + 243T^{2} \) |
| 7 | \( 1 - 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 587.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 162.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 81.3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 958.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 169.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.65e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.22e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.38e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.05854405541417134351038072232, −11.35304759743360430550470607132, −10.10427462753226056803604770707, −8.968595665529428824116130327796, −8.374126730362824721633275509994, −7.32582695781379449497196645572, −5.82193448033316566986646754901, −4.17040536299388433722684199965, −2.38888123678575678618212379598, −1.30744983143822888102858372711,
1.30744983143822888102858372711, 2.38888123678575678618212379598, 4.17040536299388433722684199965, 5.82193448033316566986646754901, 7.32582695781379449497196645572, 8.374126730362824721633275509994, 8.968595665529428824116130327796, 10.10427462753226056803604770707, 11.35304759743360430550470607132, 12.05854405541417134351038072232
