| L(s) = 1 | + 8·2-s − 26.4·3-s + 64·4-s + 139.·5-s − 211.·6-s + 458.·7-s + 512·8-s − 1.48e3·9-s + 1.11e3·10-s + 6.02e3·11-s − 1.68e3·12-s + 9.19e3·13-s + 3.67e3·14-s − 3.67e3·15-s + 4.09e3·16-s + 3.74e4·17-s − 1.19e4·18-s + 6.85e3·19-s + 8.91e3·20-s − 1.21e4·21-s + 4.82e4·22-s − 8.97e4·23-s − 1.35e4·24-s − 5.87e4·25-s + 7.35e4·26-s + 9.70e4·27-s + 2.93e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.564·3-s + 0.5·4-s + 0.498·5-s − 0.399·6-s + 0.505·7-s + 0.353·8-s − 0.681·9-s + 0.352·10-s + 1.36·11-s − 0.282·12-s + 1.16·13-s + 0.357·14-s − 0.281·15-s + 0.250·16-s + 1.84·17-s − 0.481·18-s + 0.229·19-s + 0.249·20-s − 0.285·21-s + 0.965·22-s − 1.53·23-s − 0.199·24-s − 0.751·25-s + 0.820·26-s + 0.949·27-s + 0.252·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.707173752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707173752\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 + 26.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 139.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 458.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.02e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.19e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.74e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 8.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.77e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.73e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.73e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.66e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.18e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.78e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.67e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.32e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.88e6T + 8.07e13T^{2} \) |
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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
−14.35261914732370228589503142252, −13.92006235117177666391318644723, −12.13608414343158525469942722038, −11.50192860966505691627923566581, −10.05032404959926932094125265068, −8.295448014419135264751998609793, −6.37350989514509866107127305020, −5.46608722085194530449009074179, −3.65047278078697587855394361871, −1.41991074334186519796930473037,
1.41991074334186519796930473037, 3.65047278078697587855394361871, 5.46608722085194530449009074179, 6.37350989514509866107127305020, 8.295448014419135264751998609793, 10.05032404959926932094125265068, 11.50192860966505691627923566581, 12.13608414343158525469942722038, 13.92006235117177666391318644723, 14.35261914732370228589503142252
