| L(s) = 1 | − 0.170·2-s − 0.648·3-s − 1.97·4-s + 0.110·6-s − 2.03·7-s + 0.677·8-s − 2.57·9-s − 1.88·11-s + 1.27·12-s + 4.03·13-s + 0.347·14-s + 3.82·16-s + 0.781·17-s + 0.440·18-s − 2.68·19-s + 1.32·21-s + 0.322·22-s + 5.51·23-s − 0.439·24-s − 0.689·26-s + 3.61·27-s + 4.01·28-s + 3.07·29-s + 31-s − 2.00·32-s + 1.22·33-s − 0.133·34-s + ⋯ |
| L(s) = 1 | − 0.120·2-s − 0.374·3-s − 0.985·4-s + 0.0452·6-s − 0.770·7-s + 0.239·8-s − 0.859·9-s − 0.568·11-s + 0.369·12-s + 1.11·13-s + 0.0929·14-s + 0.956·16-s + 0.189·17-s + 0.103·18-s − 0.616·19-s + 0.288·21-s + 0.0686·22-s + 1.15·23-s − 0.0897·24-s − 0.135·26-s + 0.696·27-s + 0.758·28-s + 0.571·29-s + 0.179·31-s − 0.355·32-s + 0.213·33-s − 0.0228·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7480109598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7480109598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.170T + 2T^{2} \) |
| 3 | \( 1 + 0.648T + 3T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 - 0.781T + 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 + 0.850T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.52T + 47T^{2} \) |
| 53 | \( 1 - 6.34T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6.57T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 + 3.87T + 97T^{2} \) |
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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
−10.35185697314043879139743423967, −9.361367721131540599939737678935, −8.674109801187734398276372206757, −7.990990174625013138259381168748, −6.65501387017430642984563941827, −5.83151621125190854020348486655, −5.01506764238115044229960972501, −3.86134863602753311458725211205, −2.84597905698455235248120401507, −0.73721032992548260775744495164,
0.73721032992548260775744495164, 2.84597905698455235248120401507, 3.86134863602753311458725211205, 5.01506764238115044229960972501, 5.83151621125190854020348486655, 6.65501387017430642984563941827, 7.990990174625013138259381168748, 8.674109801187734398276372206757, 9.361367721131540599939737678935, 10.35185697314043879139743423967
