| L(s) = 1 | − 24.7·3-s + 202.·5-s − 343·7-s − 1.57e3·9-s − 7.52e3·11-s − 1.70e3·13-s − 5.01e3·15-s − 2.48e3·17-s − 3.98e4·19-s + 8.49e3·21-s + 2.55e4·23-s − 3.71e4·25-s + 9.31e4·27-s + 1.48e5·29-s − 3.35e4·31-s + 1.86e5·33-s − 6.94e4·35-s + 4.00e5·37-s + 4.21e4·39-s − 3.62e5·41-s − 3.24e5·43-s − 3.18e5·45-s − 7.08e5·47-s + 1.17e5·49-s + 6.14e4·51-s − 1.85e5·53-s − 1.52e6·55-s + ⋯ |
| L(s) = 1 | − 0.529·3-s + 0.724·5-s − 0.377·7-s − 0.719·9-s − 1.70·11-s − 0.214·13-s − 0.383·15-s − 0.122·17-s − 1.33·19-s + 0.200·21-s + 0.438·23-s − 0.475·25-s + 0.910·27-s + 1.13·29-s − 0.202·31-s + 0.902·33-s − 0.273·35-s + 1.30·37-s + 0.113·39-s − 0.821·41-s − 0.622·43-s − 0.521·45-s − 0.995·47-s + 0.142·49-s + 0.0648·51-s − 0.171·53-s − 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 3 | \( 1 + 24.7T + 2.18e3T^{2} \) |
| 5 | \( 1 - 202.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.70e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.48e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.55e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.24e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.08e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.85e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.51e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.02e7T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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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
−15.11105203095668464002805776665, −13.62361943165911306943334020314, −12.62511458099424490990923546578, −11.02766854584370790829776601718, −9.968615970845589997386328613737, −8.282814878331397264335691782473, −6.36462744902072532906910942325, −5.13008884510804739598291877116, −2.57095458641617698954174958071, 0,
2.57095458641617698954174958071, 5.13008884510804739598291877116, 6.36462744902072532906910942325, 8.282814878331397264335691782473, 9.968615970845589997386328613737, 11.02766854584370790829776601718, 12.62511458099424490990923546578, 13.62361943165911306943334020314, 15.11105203095668464002805776665
