| L(s) = 1 | − 117.·2-s − 235.·3-s + 5.72e3·4-s + 2.78e4·6-s + 2.27e5·7-s + 2.91e5·8-s − 1.53e6·9-s − 7.68e5·11-s − 1.34e6·12-s + 8.24e6·13-s − 2.68e7·14-s − 8.12e7·16-s − 1.56e8·17-s + 1.81e8·18-s − 2.90e8·19-s − 5.36e7·21-s + 9.06e7·22-s + 7.93e8·23-s − 6.86e7·24-s − 9.72e8·26-s + 7.38e8·27-s + 1.30e9·28-s + 5.40e9·29-s − 1.83e9·31-s + 7.19e9·32-s + 1.81e8·33-s + 1.84e10·34-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.186·3-s + 0.698·4-s + 0.243·6-s + 0.731·7-s + 0.392·8-s − 0.965·9-s − 0.130·11-s − 0.130·12-s + 0.473·13-s − 0.953·14-s − 1.21·16-s − 1.56·17-s + 1.25·18-s − 1.41·19-s − 0.136·21-s + 0.170·22-s + 1.11·23-s − 0.0732·24-s − 0.617·26-s + 0.366·27-s + 0.511·28-s + 1.68·29-s − 0.371·31-s + 1.18·32-s + 0.0244·33-s + 2.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.7056808754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7056808754\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 117.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 235.T + 1.59e6T^{2} \) |
| 7 | \( 1 - 2.27e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.68e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 8.24e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.56e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.90e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.93e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 5.40e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.83e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.16e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 6.75e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.43e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.65e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 7.13e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 2.96e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 6.88e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.65e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.74e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 4.21e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.91e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 5.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 8.23e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 9.34e12T + 6.73e25T^{2} \) |
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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
−14.70737694742252299433994992054, −13.26053742837378565389960205624, −11.34770744707950251036060693235, −10.65785333996444492326631961779, −8.926777477133739481312818020199, −8.256957303338856991702881208139, −6.58449641743793327434933752416, −4.66280611262397693392317454731, −2.24570101778188134420839958875, −0.65802632518984988939965817206,
0.65802632518984988939965817206, 2.24570101778188134420839958875, 4.66280611262397693392317454731, 6.58449641743793327434933752416, 8.256957303338856991702881208139, 8.926777477133739481312818020199, 10.65785333996444492326631961779, 11.34770744707950251036060693235, 13.26053742837378565389960205624, 14.70737694742252299433994992054
