| L(s) = 1 | + 128·2-s − 2.18e3·3-s + 1.63e4·4-s + 2.40e5·5-s − 2.79e5·6-s − 3.60e6·7-s + 2.09e6·8-s + 4.78e6·9-s + 3.07e7·10-s + 6.90e7·11-s − 3.58e7·12-s − 6.27e7·13-s − 4.61e8·14-s − 5.25e8·15-s + 2.68e8·16-s − 1.17e9·17-s + 6.12e8·18-s − 3.66e9·19-s + 3.93e9·20-s + 7.88e9·21-s + 8.83e9·22-s + 8.94e9·23-s − 4.58e9·24-s + 2.73e10·25-s − 8.03e9·26-s − 1.04e10·27-s − 5.90e10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.37·5-s − 0.408·6-s − 1.65·7-s + 0.353·8-s + 0.333·9-s + 0.973·10-s + 1.06·11-s − 0.288·12-s − 0.277·13-s − 1.16·14-s − 0.794·15-s + 0.250·16-s − 0.692·17-s + 0.235·18-s − 0.941·19-s + 0.688·20-s + 0.954·21-s + 0.755·22-s + 0.547·23-s − 0.204·24-s + 0.894·25-s − 0.196·26-s − 0.192·27-s − 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 3 | \( 1 + 2.18e3T \) |
| 13 | \( 1 + 6.27e7T \) |
| good | 5 | \( 1 - 2.40e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 3.60e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 6.90e7T + 4.17e15T^{2} \) |
| 17 | \( 1 + 1.17e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 3.66e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 8.94e9T + 2.66e20T^{2} \) |
| 29 | \( 1 - 4.21e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.28e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.66e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 5.96e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 5.90e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 1.83e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 9.23e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.78e11T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.39e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 6.13e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.60e12T + 5.87e27T^{2} \) |
| 73 | \( 1 - 5.54e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 3.36e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 4.68e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.79e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.07e15T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90690226002028358068945961544, −9.875416678429204384596079732853, −9.103302801399212969440252005179, −6.70411708570700162200258839685, −6.45309744777023932071894266380, −5.38538531126698181134583774763, −3.99139077129569690538618227124, −2.70519170074026096922388321332, −1.53037528606470010977617359207, 0,
1.53037528606470010977617359207, 2.70519170074026096922388321332, 3.99139077129569690538618227124, 5.38538531126698181134583774763, 6.45309744777023932071894266380, 6.70411708570700162200258839685, 9.103302801399212969440252005179, 9.875416678429204384596079732853, 10.90690226002028358068945961544
