| L(s) = 1 | + 4·2-s − 9.29·3-s + 16·4-s − 44.4·5-s − 37.1·6-s + 69.2·7-s + 64·8-s − 156.·9-s − 177.·10-s − 705.·11-s − 148.·12-s − 458.·13-s + 277.·14-s + 412.·15-s + 256·16-s + 132.·17-s − 626.·18-s − 2.15e3·19-s − 710.·20-s − 643.·21-s − 2.82e3·22-s + 4.03e3·23-s − 594.·24-s − 1.15e3·25-s − 1.83e3·26-s + 3.71e3·27-s + 1.10e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.596·3-s + 0.5·4-s − 0.794·5-s − 0.421·6-s + 0.534·7-s + 0.353·8-s − 0.644·9-s − 0.561·10-s − 1.75·11-s − 0.298·12-s − 0.753·13-s + 0.377·14-s + 0.473·15-s + 0.250·16-s + 0.110·17-s − 0.455·18-s − 1.36·19-s − 0.397·20-s − 0.318·21-s − 1.24·22-s + 1.58·23-s − 0.210·24-s − 0.368·25-s − 0.532·26-s + 0.980·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 4T \) |
| 31 | \( 1 - 961T \) |
| good | 3 | \( 1 + 9.29T + 243T^{2} \) |
| 5 | \( 1 + 44.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 69.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 705.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 458.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 132.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.30e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 35.0T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.51e4T + 8.58e9T^{2} \) |
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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
−13.29022780556621940214288255340, −12.31261097793900319912831373497, −11.27572538225196273679207887928, −10.52496135867989714765836351273, −8.385635796920390091771101838873, −7.27945109202843231950787448482, −5.59887890438818739280333659326, −4.61165891140142596771660924265, −2.70635000532300116515577256862, 0,
2.70635000532300116515577256862, 4.61165891140142596771660924265, 5.59887890438818739280333659326, 7.27945109202843231950787448482, 8.385635796920390091771101838873, 10.52496135867989714765836351273, 11.27572538225196273679207887928, 12.31261097793900319912831373497, 13.29022780556621940214288255340
