| L(s) = 1 | − 0.260·2-s + 1.85·3-s − 1.93·4-s − 0.484·6-s + 1.27·7-s + 1.02·8-s + 0.456·9-s − 5.63·11-s − 3.59·12-s + 6.25·13-s − 0.332·14-s + 3.59·16-s − 0.195·17-s − 0.119·18-s + 3.74·19-s + 2.36·21-s + 1.46·22-s − 8.06·23-s + 1.90·24-s − 1.62·26-s − 4.72·27-s − 2.46·28-s + 6.42·29-s − 10.3·31-s − 2.98·32-s − 10.4·33-s + 0.0509·34-s + ⋯ |
| L(s) = 1 | − 0.184·2-s + 1.07·3-s − 0.966·4-s − 0.197·6-s + 0.481·7-s + 0.362·8-s + 0.152·9-s − 1.69·11-s − 1.03·12-s + 1.73·13-s − 0.0887·14-s + 0.899·16-s − 0.0473·17-s − 0.0280·18-s + 0.860·19-s + 0.517·21-s + 0.313·22-s − 1.68·23-s + 0.388·24-s − 0.319·26-s − 0.909·27-s − 0.465·28-s + 1.19·29-s − 1.85·31-s − 0.527·32-s − 1.82·33-s + 0.00873·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.260T + 2T^{2} \) |
| 3 | \( 1 - 1.85T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 + 0.195T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + 8.06T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 4.71T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 - 0.879T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.908003056106723008733536575908, −7.54053354479904315035835092058, −6.15386541713505325111362838994, −5.48078267718444366789309761555, −4.83464670855234058102048278193, −3.78172141927674668200069613622, −3.39625118244798277809389766339, −2.34853288459731207747832273045, −1.39922817677304063831944930829, 0,
1.39922817677304063831944930829, 2.34853288459731207747832273045, 3.39625118244798277809389766339, 3.78172141927674668200069613622, 4.83464670855234058102048278193, 5.48078267718444366789309761555, 6.15386541713505325111362838994, 7.54053354479904315035835092058, 7.908003056106723008733536575908
