| L(s) = 1 | − 8·2-s − 54.3·3-s + 64·4-s + 58.6·5-s + 434.·6-s − 343·7-s − 512·8-s + 768.·9-s − 469.·10-s + 7.91e3·11-s − 3.47e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 3.19e3·15-s + 4.09e3·16-s + 1.09e4·17-s − 6.14e3·18-s − 5.49e4·19-s + 3.75e3·20-s + 1.86e4·21-s − 6.33e4·22-s + 8.52e4·23-s + 2.78e4·24-s − 7.46e4·25-s + 1.75e4·26-s + 7.71e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.16·3-s + 0.5·4-s + 0.209·5-s + 0.821·6-s − 0.377·7-s − 0.353·8-s + 0.351·9-s − 0.148·10-s + 1.79·11-s − 0.581·12-s − 0.277·13-s + 0.267·14-s − 0.244·15-s + 0.250·16-s + 0.538·17-s − 0.248·18-s − 1.83·19-s + 0.104·20-s + 0.439·21-s − 1.26·22-s + 1.46·23-s + 0.410·24-s − 0.955·25-s + 0.196·26-s + 0.754·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7538627070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7538627070\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 + 54.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 58.6T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.91e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.77e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.58e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.97e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.00e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.20e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.81e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.60e7T + 8.07e13T^{2} \) |
| 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
−11.30503594543435089039651107261, −10.41467247310341349346913486081, −9.408837195099040759705570093831, −8.503863903663273939890405616089, −6.81793392356389685340116355818, −6.41708992425533268045296008155, −5.17276781664709027924382103761, −3.64842477801475137341018388483, −1.80191257466579686118399834020, −0.56185772272962849032208746990,
0.56185772272962849032208746990, 1.80191257466579686118399834020, 3.64842477801475137341018388483, 5.17276781664709027924382103761, 6.41708992425533268045296008155, 6.81793392356389685340116355818, 8.503863903663273939890405616089, 9.408837195099040759705570093831, 10.41467247310341349346913486081, 11.30503594543435089039651107261
