| L(s) = 1 | + 7.33·2-s + 21.7·4-s + 0.408·5-s − 4.34·7-s − 74.9·8-s + 2.99·10-s − 428.·11-s + 76.4·13-s − 31.8·14-s − 1.24e3·16-s + 71.8·17-s − 1.73e3·19-s + 8.89·20-s − 3.14e3·22-s + 529·23-s − 3.12e3·25-s + 560.·26-s − 94.5·28-s + 1.66e3·29-s − 3.62e3·31-s − 6.74e3·32-s + 526.·34-s − 1.77·35-s − 1.46e3·37-s − 1.27e4·38-s − 30.6·40-s − 7.59e3·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.680·4-s + 0.00730·5-s − 0.0335·7-s − 0.414·8-s + 0.00947·10-s − 1.06·11-s + 0.125·13-s − 0.0434·14-s − 1.21·16-s + 0.0602·17-s − 1.10·19-s + 0.00497·20-s − 1.38·22-s + 0.208·23-s − 0.999·25-s + 0.162·26-s − 0.0227·28-s + 0.367·29-s − 0.676·31-s − 1.16·32-s + 0.0781·34-s − 0.000244·35-s − 0.176·37-s − 1.42·38-s − 0.00302·40-s − 0.705·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 7.33T + 32T^{2} \) |
| 5 | \( 1 - 0.408T + 3.12e3T^{2} \) |
| 7 | \( 1 + 4.34T + 1.68e4T^{2} \) |
| 11 | \( 1 + 428.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 76.4T + 3.71e5T^{2} \) |
| 17 | \( 1 - 71.8T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.73e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.59e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.93e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.09e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 411.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
| 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
−11.26755411261011188753812195765, −10.28690846066836724323274981902, −9.011339266717079063769628780160, −7.85825370497309893225649480097, −6.53260982611786677188943917138, −5.55811974953581805464758799789, −4.59811355231177093709013402428, −3.45577366272763574759939968600, −2.22772427764659088478050371537, 0,
2.22772427764659088478050371537, 3.45577366272763574759939968600, 4.59811355231177093709013402428, 5.55811974953581805464758799789, 6.53260982611786677188943917138, 7.85825370497309893225649480097, 9.011339266717079063769628780160, 10.28690846066836724323274981902, 11.26755411261011188753812195765
