| L(s) = 1 | + 2-s − 0.917·3-s + 4-s − 0.382·5-s − 0.917·6-s + 1.33·7-s + 8-s − 2.15·9-s − 0.382·10-s − 2.71·11-s − 0.917·12-s − 2.03·13-s + 1.33·14-s + 0.350·15-s + 16-s + 0.649·17-s − 2.15·18-s + 5.96·19-s − 0.382·20-s − 1.22·21-s − 2.71·22-s − 0.117·23-s − 0.917·24-s − 4.85·25-s − 2.03·26-s + 4.73·27-s + 1.33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.529·3-s + 0.5·4-s − 0.170·5-s − 0.374·6-s + 0.505·7-s + 0.353·8-s − 0.719·9-s − 0.120·10-s − 0.819·11-s − 0.264·12-s − 0.565·13-s + 0.357·14-s + 0.0905·15-s + 0.250·16-s + 0.157·17-s − 0.508·18-s + 1.36·19-s − 0.0854·20-s − 0.267·21-s − 0.579·22-s − 0.0244·23-s − 0.187·24-s − 0.970·25-s − 0.399·26-s + 0.910·27-s + 0.252·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 0.917T + 3T^{2} \) |
| 5 | \( 1 + 0.382T + 5T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 0.649T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 + 0.117T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 1.67T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.985899299872732689014195828165, −7.30431703675896371802417137709, −6.46392164563328148985142289491, −5.61624811951675439494431446981, −5.14011794620683168085496076696, −4.53242353363215042154390652011, −3.32523905908493759631229052443, −2.71837236918067428926445477414, −1.51015426976798624698189186171, 0,
1.51015426976798624698189186171, 2.71837236918067428926445477414, 3.32523905908493759631229052443, 4.53242353363215042154390652011, 5.14011794620683168085496076696, 5.61624811951675439494431446981, 6.46392164563328148985142289491, 7.30431703675896371802417137709, 7.985899299872732689014195828165
