| L(s) = 1 | + 1.05·2-s + 3-s − 0.877·4-s + 3.52·5-s + 1.05·6-s − 3.11·7-s − 3.04·8-s + 9-s + 3.73·10-s − 0.0526·11-s − 0.877·12-s − 5.31·13-s − 3.29·14-s + 3.52·15-s − 1.47·16-s + 17-s + 1.05·18-s + 7.51·19-s − 3.09·20-s − 3.11·21-s − 0.0558·22-s − 2.09·23-s − 3.04·24-s + 7.43·25-s − 5.63·26-s + 27-s + 2.73·28-s + ⋯ |
| L(s) = 1 | + 0.749·2-s + 0.577·3-s − 0.438·4-s + 1.57·5-s + 0.432·6-s − 1.17·7-s − 1.07·8-s + 0.333·9-s + 1.18·10-s − 0.0158·11-s − 0.253·12-s − 1.47·13-s − 0.881·14-s + 0.910·15-s − 0.368·16-s + 0.242·17-s + 0.249·18-s + 1.72·19-s − 0.691·20-s − 0.678·21-s − 0.0118·22-s − 0.437·23-s − 0.622·24-s + 1.48·25-s − 1.10·26-s + 0.192·27-s + 0.515·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 0.0526T + 11T^{2} \) |
| 13 | \( 1 + 5.31T + 13T^{2} \) |
| 19 | \( 1 - 7.51T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 + 9.36T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 0.00547T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−7.22191649626855532266488727605, −6.74122810017404915003577936517, −5.91186550667564097484950717687, −5.30574731932127897910090201629, −4.91500400564037707012802244227, −3.72313448591058934039852873456, −3.10643911847827810467905229229, −2.54254767090864901980854329142, −1.52895287629342556844446838397, 0,
1.52895287629342556844446838397, 2.54254767090864901980854329142, 3.10643911847827810467905229229, 3.72313448591058934039852873456, 4.91500400564037707012802244227, 5.30574731932127897910090201629, 5.91186550667564097484950717687, 6.74122810017404915003577936517, 7.22191649626855532266488727605
