| L(s) = 1 | + 66.7·2-s + 729·3-s − 3.73e3·4-s + 3.60e4·5-s + 4.86e4·6-s − 2.69e5·7-s − 7.96e5·8-s + 5.31e5·9-s + 2.40e6·10-s + 1.04e7·11-s − 2.71e6·12-s − 7.55e6·13-s − 1.80e7·14-s + 2.62e7·15-s − 2.26e7·16-s − 1.33e8·17-s + 3.54e7·18-s + 2.81e8·19-s − 1.34e8·20-s − 1.96e8·21-s + 6.94e8·22-s + 2.81e8·23-s − 5.80e8·24-s + 8.07e7·25-s − 5.04e8·26-s + 3.87e8·27-s + 1.00e9·28-s + ⋯ |
| L(s) = 1 | + 0.738·2-s + 0.577·3-s − 0.455·4-s + 1.03·5-s + 0.426·6-s − 0.866·7-s − 1.07·8-s + 0.333·9-s + 0.762·10-s + 1.77·11-s − 0.262·12-s − 0.434·13-s − 0.639·14-s + 0.596·15-s − 0.337·16-s − 1.33·17-s + 0.246·18-s + 1.37·19-s − 0.470·20-s − 0.500·21-s + 1.30·22-s + 0.397·23-s − 0.620·24-s + 0.0661·25-s − 0.320·26-s + 0.192·27-s + 0.394·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(4.173153114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.173153114\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 66.7T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.60e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.69e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.04e7T + 3.45e13T^{2} \) |
| 13 | \( 1 + 7.55e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.33e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.81e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.81e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.19e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.46e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 7.12e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.84e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.95e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.09e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.87e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.11e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 9.50e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 2.78e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 5.64e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.36e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.62e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.86e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 8.83e12T + 6.73e25T^{2} \) |
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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
−9.912217947115501011382209601567, −9.355336376987430978168375880173, −8.769409232648173950624326153065, −6.94714291183350628070367887970, −6.24449197528836492981130606512, −5.13233111938152133369004841920, −3.99333138043340226405600970395, −3.17301361576883420917188101207, −2.03202068427665547557624912273, −0.75699411441656317557277869526,
0.75699411441656317557277869526, 2.03202068427665547557624912273, 3.17301361576883420917188101207, 3.99333138043340226405600970395, 5.13233111938152133369004841920, 6.24449197528836492981130606512, 6.94714291183350628070367887970, 8.769409232648173950624326153065, 9.355336376987430978168375880173, 9.912217947115501011382209601567
