| L(s) = 1 | + 8·2-s + 22.3·3-s + 64·4-s − 504.·5-s + 179.·6-s + 892.·7-s + 512·8-s − 1.68e3·9-s − 4.03e3·10-s + 4.70e3·11-s + 1.43e3·12-s − 1.18e4·13-s + 7.13e3·14-s − 1.12e4·15-s + 4.09e3·16-s − 2.26e4·17-s − 1.34e4·18-s − 5.16e4·19-s − 3.22e4·20-s + 1.99e4·21-s + 3.76e4·22-s − 7.28e4·23-s + 1.14e4·24-s + 1.76e5·25-s − 9.47e4·26-s − 8.67e4·27-s + 5.71e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.478·3-s + 0.5·4-s − 1.80·5-s + 0.338·6-s + 0.983·7-s + 0.353·8-s − 0.770·9-s − 1.27·10-s + 1.06·11-s + 0.239·12-s − 1.49·13-s + 0.695·14-s − 0.864·15-s + 0.250·16-s − 1.11·17-s − 0.545·18-s − 1.72·19-s − 0.902·20-s + 0.470·21-s + 0.753·22-s − 1.24·23-s + 0.169·24-s + 2.25·25-s − 1.05·26-s − 0.847·27-s + 0.491·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 37 | \( 1 + 5.06e4T \) |
| good | 3 | \( 1 - 22.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 504.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 892.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.70e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.18e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.28e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.62e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.03e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + 5.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.52e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.33e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.02e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.25e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.24e6T + 8.07e13T^{2} \) |
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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
−12.22708014114808240083355943309, −11.79871004670638855411458284575, −10.79091518162117528639326852917, −8.700110102557052583373379688522, −7.948327331037264779109150561429, −6.72621037937633725945670666169, −4.68917329768906413932991237955, −3.93188502286257509403606520755, −2.32172408316003384122020943012, 0,
2.32172408316003384122020943012, 3.93188502286257509403606520755, 4.68917329768906413932991237955, 6.72621037937633725945670666169, 7.948327331037264779109150561429, 8.700110102557052583373379688522, 10.79091518162117528639326852917, 11.79871004670638855411458284575, 12.22708014114808240083355943309
