| L(s) = 1 | − 83.3·2-s − 243·3-s + 4.90e3·4-s − 4.95e3·5-s + 2.02e4·6-s + 1.43e4·7-s − 2.38e5·8-s + 5.90e4·9-s + 4.13e5·10-s − 5.46e5·11-s − 1.19e6·12-s + 3.49e5·13-s − 1.19e6·14-s + 1.20e6·15-s + 9.83e6·16-s − 7.20e6·17-s − 4.92e6·18-s − 6.45e6·19-s − 2.43e7·20-s − 3.47e6·21-s + 4.55e7·22-s − 3.67e7·23-s + 5.79e7·24-s − 2.42e7·25-s − 2.91e7·26-s − 1.43e7·27-s + 7.01e7·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.39·4-s − 0.709·5-s + 1.06·6-s + 0.321·7-s − 2.57·8-s + 0.333·9-s + 1.30·10-s − 1.02·11-s − 1.38·12-s + 0.261·13-s − 0.592·14-s + 0.409·15-s + 2.34·16-s − 1.23·17-s − 0.614·18-s − 0.597·19-s − 1.70·20-s − 0.185·21-s + 1.88·22-s − 1.18·23-s + 1.48·24-s − 0.496·25-s − 0.481·26-s − 0.192·27-s + 0.770·28-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
| \(L(6)\) |
\(\approx\) |
\(0.03177217403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03177217403\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 83.3T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.95e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.43e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.46e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 3.49e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.20e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 6.45e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.67e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.87e6T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.50e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.68e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.73e6T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.13e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + 9.01e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.88e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.74e8T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.65e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.97e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.79e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.92e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.40e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.27e10T + 7.15e21T^{2} \) |
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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
−10.61342984158865008578927007165, −9.727761861638845646894489586661, −8.483715473096418483948335902190, −7.926910296297469480657295598129, −6.94930275234760610680856371099, −5.90863395600798758744927379750, −4.30461736691449405123966649607, −2.56868627585603782173395612448, −1.51904800349624438382167434832, −0.11044563945977581989846860035,
0.11044563945977581989846860035, 1.51904800349624438382167434832, 2.56868627585603782173395612448, 4.30461736691449405123966649607, 5.90863395600798758744927379750, 6.94930275234760610680856371099, 7.926910296297469480657295598129, 8.483715473096418483948335902190, 9.727761861638845646894489586661, 10.61342984158865008578927007165
