| L(s) = 1 | − 0.134·2-s − 0.934·3-s − 1.98·4-s + 5-s + 0.125·6-s − 1.20·7-s + 0.535·8-s − 2.12·9-s − 0.134·10-s + 2.09·11-s + 1.85·12-s + 2.95·13-s + 0.162·14-s − 0.934·15-s + 3.89·16-s − 17-s + 0.285·18-s − 5.68·19-s − 1.98·20-s + 1.12·21-s − 0.282·22-s − 2.33·23-s − 0.500·24-s + 25-s − 0.396·26-s + 4.79·27-s + 2.38·28-s + ⋯ |
| L(s) = 1 | − 0.0950·2-s − 0.539·3-s − 0.990·4-s + 0.447·5-s + 0.0513·6-s − 0.455·7-s + 0.189·8-s − 0.708·9-s − 0.0425·10-s + 0.632·11-s + 0.534·12-s + 0.818·13-s + 0.0433·14-s − 0.241·15-s + 0.972·16-s − 0.242·17-s + 0.0673·18-s − 1.30·19-s − 0.443·20-s + 0.245·21-s − 0.0601·22-s − 0.485·23-s − 0.102·24-s + 0.200·25-s − 0.0778·26-s + 0.922·27-s + 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 0.134T + 2T^{2} \) |
| 3 | \( 1 + 0.934T + 3T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 1.31T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 0.797T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 4.26T + 97T^{2} \) |
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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
−8.003691689094319405791108539837, −6.59407587962400764185516043083, −6.39770593581478050746320923748, −5.59289998588363080412808956796, −4.90732355899246679138066261015, −4.06483201661588569103735521907, −3.41400524197082123740705372336, −2.26505175358021631629459406635, −1.06016693836778462526494807332, 0,
1.06016693836778462526494807332, 2.26505175358021631629459406635, 3.41400524197082123740705372336, 4.06483201661588569103735521907, 4.90732355899246679138066261015, 5.59289998588363080412808956796, 6.39770593581478050746320923748, 6.59407587962400764185516043083, 8.003691689094319405791108539837
