| L(s) = 1 | + 8·2-s + 64·4-s + 235.·5-s − 1.11e3·7-s + 512·8-s + 1.88e3·10-s − 1.42e3·11-s − 5.51e3·13-s − 8.89e3·14-s + 4.09e3·16-s + 2.74e4·17-s + 6.85e3·19-s + 1.50e4·20-s − 1.14e4·22-s + 6.22e4·23-s − 2.26e4·25-s − 4.41e4·26-s − 7.11e4·28-s − 2.09e5·29-s − 1.30e5·31-s + 3.27e4·32-s + 2.19e5·34-s − 2.61e5·35-s − 3.54e4·37-s + 5.48e4·38-s + 1.20e5·40-s − 5.63e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.842·5-s − 1.22·7-s + 0.353·8-s + 0.595·10-s − 0.323·11-s − 0.696·13-s − 0.866·14-s + 0.250·16-s + 1.35·17-s + 0.229·19-s + 0.421·20-s − 0.228·22-s + 1.06·23-s − 0.290·25-s − 0.492·26-s − 0.612·28-s − 1.59·29-s − 0.783·31-s + 0.176·32-s + 0.958·34-s − 1.03·35-s − 0.115·37-s + 0.162·38-s + 0.297·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 5 | \( 1 - 235.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.51e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.74e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 6.22e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.30e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.54e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.63e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.84e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.84e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.65e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.49e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.96e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.52e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.35e7T + 8.07e13T^{2} \) |
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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
−9.835807647188720063531787005563, −9.275849175293502066624786458372, −7.68182272960311616031251296567, −6.81888502461990976597691487458, −5.78628916548815075498265023957, −5.17134327838798145846116750800, −3.61426356082939387735492954907, −2.82261077660054310123664428802, −1.59359137848107512366772378634, 0,
1.59359137848107512366772378634, 2.82261077660054310123664428802, 3.61426356082939387735492954907, 5.17134327838798145846116750800, 5.78628916548815075498265023957, 6.81888502461990976597691487458, 7.68182272960311616031251296567, 9.275849175293502066624786458372, 9.835807647188720063531787005563
