| L(s) = 1 | + 4.09e3·2-s + 1.08e6·3-s + 1.67e7·4-s + 4.45e9·6-s − 4.89e10·7-s + 6.87e10·8-s + 3.34e11·9-s + 1.14e13·11-s + 1.82e13·12-s + 1.28e14·13-s − 2.00e14·14-s + 2.81e14·16-s − 3.78e15·17-s + 1.36e15·18-s − 1.02e16·19-s − 5.32e16·21-s + 4.69e16·22-s − 1.81e17·23-s + 7.46e16·24-s + 5.24e17·26-s − 5.57e17·27-s − 8.21e17·28-s + 2.19e18·29-s + 2.89e18·31-s + 1.15e18·32-s + 1.24e19·33-s − 1.55e19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s + 0.835·6-s − 1.33·7-s + 0.353·8-s + 0.394·9-s + 1.10·11-s + 0.590·12-s + 1.52·13-s − 0.945·14-s + 0.250·16-s − 1.57·17-s + 0.278·18-s − 1.05·19-s − 1.57·21-s + 0.778·22-s − 1.73·23-s + 0.417·24-s + 1.07·26-s − 0.715·27-s − 0.668·28-s + 1.15·29-s + 0.660·31-s + 0.176·32-s + 1.29·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.08e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 4.89e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.14e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.28e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.78e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.02e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.81e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 2.19e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 2.89e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.56e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 4.22e18T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.22e19T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.60e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 2.54e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 2.20e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 7.41e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 6.01e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 3.59e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.98e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 7.84e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 4.60e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 5.94e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 8.71e24T + 4.66e49T^{2} \) |
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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
−10.23931853473784408435214627933, −9.016073984361160645479626715148, −8.343048018129732200863374770503, −6.58352243489922498727409615351, −6.23885776401801481471348580652, −4.17525603782516464402099278373, −3.63739006076431670839858848036, −2.62855393423141583312383994949, −1.62351563379580516848345803221, 0,
1.62351563379580516848345803221, 2.62855393423141583312383994949, 3.63739006076431670839858848036, 4.17525603782516464402099278373, 6.23885776401801481471348580652, 6.58352243489922498727409615351, 8.343048018129732200863374770503, 9.016073984361160645479626715148, 10.23931853473784408435214627933
