| L(s) = 1 | − 8.09·2-s − 62.4·4-s − 308.·5-s + 97.1·7-s + 1.54e3·8-s + 2.49e3·10-s + 1.33e3·11-s + 1.28e4·13-s − 787.·14-s − 4.49e3·16-s − 8.78e3·17-s − 1.14e3·19-s + 1.92e4·20-s − 1.07e4·22-s − 2.30e4·23-s + 1.67e4·25-s − 1.03e5·26-s − 6.06e3·28-s − 3.82e4·29-s + 2.54e5·31-s − 1.60e5·32-s + 7.11e4·34-s − 2.99e4·35-s + 4.63e5·37-s + 9.29e3·38-s − 4.75e5·40-s − 6.29e5·41-s + ⋯ |
| L(s) = 1 | − 0.715·2-s − 0.487·4-s − 1.10·5-s + 0.107·7-s + 1.06·8-s + 0.788·10-s + 0.301·11-s + 1.61·13-s − 0.0766·14-s − 0.274·16-s − 0.433·17-s − 0.0383·19-s + 0.537·20-s − 0.215·22-s − 0.394·23-s + 0.214·25-s − 1.15·26-s − 0.0522·28-s − 0.291·29-s + 1.53·31-s − 0.868·32-s + 0.310·34-s − 0.118·35-s + 1.50·37-s + 0.0274·38-s − 1.17·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 + 8.09T + 128T^{2} \) |
| 5 | \( 1 + 308.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 97.1T + 8.23e5T^{2} \) |
| 13 | \( 1 - 1.28e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.14e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.63e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.41e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.40e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.30e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.59e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.43e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.00e6T + 8.07e13T^{2} \) |
| 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
−11.70053187297078405520503199839, −10.91011845543542553152605059263, −9.659065981229895764520639692662, −8.457818044049742696248694047667, −7.953790113286285457408574931026, −6.43451495163065716693404172060, −4.60983452581569709974802590004, −3.60414826268537065768120760029, −1.30087833475015077456762176231, 0,
1.30087833475015077456762176231, 3.60414826268537065768120760029, 4.60983452581569709974802590004, 6.43451495163065716693404172060, 7.953790113286285457408574931026, 8.457818044049742696248694047667, 9.659065981229895764520639692662, 10.91011845543542553152605059263, 11.70053187297078405520503199839
