| L(s) = 1 | + 2-s + 0.523·3-s + 4-s + 2.72·5-s + 0.523·6-s − 4.67·7-s + 8-s − 2.72·9-s + 2.72·10-s − 11-s + 0.523·12-s + 2.67·13-s − 4.67·14-s + 1.42·15-s + 16-s + 0.201·17-s − 2.72·18-s + 2.72·20-s − 2.45·21-s − 22-s − 1.79·23-s + 0.523·24-s + 2.42·25-s + 2.67·26-s − 3·27-s − 4.67·28-s + 4.92·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.302·3-s + 0.5·4-s + 1.21·5-s + 0.213·6-s − 1.76·7-s + 0.353·8-s − 0.908·9-s + 0.861·10-s − 0.301·11-s + 0.151·12-s + 0.742·13-s − 1.25·14-s + 0.368·15-s + 0.250·16-s + 0.0488·17-s − 0.642·18-s + 0.609·20-s − 0.534·21-s − 0.213·22-s − 0.375·23-s + 0.106·24-s + 0.485·25-s + 0.525·26-s − 0.577·27-s − 0.883·28-s + 0.914·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.523T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 0.201T + 17T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 5.45T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.18900046192198640125843380431, −6.51167339999349259858063176736, −6.08944499686154071586968192250, −5.61347533713577982534564988568, −4.78140971530590111503041636912, −3.60399909123137019320919707806, −3.13967765210225317753310118247, −2.52798494532862335140905539990, −1.55916907006497981232321555379, 0,
1.55916907006497981232321555379, 2.52798494532862335140905539990, 3.13967765210225317753310118247, 3.60399909123137019320919707806, 4.78140971530590111503041636912, 5.61347533713577982534564988568, 6.08944499686154071586968192250, 6.51167339999349259858063176736, 7.18900046192198640125843380431
