| L(s) = 1 | − 44.5·2-s + 243·3-s − 66.0·4-s − 2.49e3·5-s − 1.08e4·6-s − 1.02e4·7-s + 9.41e4·8-s + 5.90e4·9-s + 1.10e5·10-s − 9.35e5·11-s − 1.60e4·12-s − 1.88e6·13-s + 4.54e5·14-s − 6.05e5·15-s − 4.05e6·16-s − 7.73e6·17-s − 2.62e6·18-s + 2.90e6·19-s + 1.64e5·20-s − 2.48e6·21-s + 4.16e7·22-s + 3.68e7·23-s + 2.28e7·24-s − 4.26e7·25-s + 8.37e7·26-s + 1.43e7·27-s + 6.74e5·28-s + ⋯ |
| L(s) = 1 | − 0.983·2-s + 0.577·3-s − 0.0322·4-s − 0.356·5-s − 0.567·6-s − 0.229·7-s + 1.01·8-s + 0.333·9-s + 0.350·10-s − 1.75·11-s − 0.0186·12-s − 1.40·13-s + 0.226·14-s − 0.205·15-s − 0.966·16-s − 1.32·17-s − 0.327·18-s + 0.268·19-s + 0.0114·20-s − 0.132·21-s + 1.72·22-s + 1.19·23-s + 0.586·24-s − 0.872·25-s + 1.38·26-s + 0.192·27-s + 0.00741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.1284898606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1284898606\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 44.5T + 2.04e3T^{2} \) |
| 5 | \( 1 + 2.49e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.02e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.35e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.88e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.73e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.90e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.68e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.85e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 9.30e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.96e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.54e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.17e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 3.82e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.18e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 3.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.49e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.28e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.82e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.69e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.42e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.60e10T + 7.15e21T^{2} \) |
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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
−10.39340010708831607510350034951, −9.520139864970083813799986829329, −8.763549612561781711276533184252, −7.60900009343842515041159922644, −7.30723899401759976371977611685, −5.30913511259395115674264508257, −4.33240889102086914786812269309, −2.83764870709683681243653969659, −1.87612068897408924183958315790, −0.17417578859974240725981666721,
0.17417578859974240725981666721, 1.87612068897408924183958315790, 2.83764870709683681243653969659, 4.33240889102086914786812269309, 5.30913511259395115674264508257, 7.30723899401759976371977611685, 7.60900009343842515041159922644, 8.763549612561781711276533184252, 9.520139864970083813799986829329, 10.39340010708831607510350034951
