| L(s) = 1 | − 1.64·2-s − 3-s + 0.696·4-s + 5-s + 1.64·6-s − 2.05·7-s + 2.14·8-s + 9-s − 1.64·10-s + 4.93·11-s − 0.696·12-s + 3.82·13-s + 3.37·14-s − 15-s − 4.90·16-s − 1.93·17-s − 1.64·18-s − 1.25·19-s + 0.696·20-s + 2.05·21-s − 8.10·22-s + 6.51·23-s − 2.14·24-s + 25-s − 6.27·26-s − 27-s − 1.43·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.348·4-s + 0.447·5-s + 0.670·6-s − 0.776·7-s + 0.756·8-s + 0.333·9-s − 0.519·10-s + 1.48·11-s − 0.201·12-s + 1.05·13-s + 0.901·14-s − 0.258·15-s − 1.22·16-s − 0.468·17-s − 0.387·18-s − 0.286·19-s + 0.155·20-s + 0.448·21-s − 1.72·22-s + 1.35·23-s − 0.436·24-s + 0.200·25-s − 1.23·26-s − 0.192·27-s − 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 7.99T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 6.90T + 79T^{2} \) |
| 83 | \( 1 - 9.69T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−7.68883694717340150486736299317, −7.03908672586154380226402753330, −6.39980853726622766995379045846, −5.91266050232767522575267250172, −4.83671965615118621874593620789, −4.03694762027227174059441787229, −3.23204323157650219199676732705, −1.77653169192587625386890174314, −1.19353179479555200233635329116, 0,
1.19353179479555200233635329116, 1.77653169192587625386890174314, 3.23204323157650219199676732705, 4.03694762027227174059441787229, 4.83671965615118621874593620789, 5.91266050232767522575267250172, 6.39980853726622766995379045846, 7.03908672586154380226402753330, 7.68883694717340150486736299317
