| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 25·5-s − 36·6-s + 179.·7-s + 64·8-s + 81·9-s − 100·10-s + 489.·11-s − 144·12-s + 752.·13-s + 716.·14-s + 225·15-s + 256·16-s − 747.·17-s + 324·18-s − 361·19-s − 400·20-s − 1.61e3·21-s + 1.95e3·22-s + 165.·23-s − 576·24-s + 625·25-s + 3.01e3·26-s − 729·27-s + 2.86e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.38·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.21·11-s − 0.288·12-s + 1.23·13-s + 0.977·14-s + 0.258·15-s + 0.250·16-s − 0.627·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.798·21-s + 0.861·22-s + 0.0654·23-s − 0.204·24-s + 0.200·25-s + 0.873·26-s − 0.192·27-s + 0.691·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.792617258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.792617258\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 - 179.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 489.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 752.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 747.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 165.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 825.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.52e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.95e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.37e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.22e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.06e4T + 8.58e9T^{2} \) |
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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
−10.36578379604870137410327920852, −8.822239691857908196980563925851, −8.222479311381969304188694779702, −6.97343323431033672917035571607, −6.31778387496329852020712310212, −5.16902389070661309259401303871, −4.39278728738939797405143304719, −3.57735745621786579966845452861, −1.86533716488006817633486049521, −0.961948812964953341637845571430,
0.961948812964953341637845571430, 1.86533716488006817633486049521, 3.57735745621786579966845452861, 4.39278728738939797405143304719, 5.16902389070661309259401303871, 6.31778387496329852020712310212, 6.97343323431033672917035571607, 8.222479311381969304188694779702, 8.822239691857908196980563925851, 10.36578379604870137410327920852
