| L(s) = 1 | + 40.1·2-s − 243·3-s − 437.·4-s + 1.08e4·5-s − 9.75e3·6-s − 2.15e3·7-s − 9.97e4·8-s + 5.90e4·9-s + 4.35e5·10-s − 5.58e5·11-s + 1.06e5·12-s + 1.91e6·13-s − 8.65e4·14-s − 2.63e6·15-s − 3.10e6·16-s + 3.75e6·17-s + 2.36e6·18-s − 1.12e7·19-s − 4.75e6·20-s + 5.24e5·21-s − 2.23e7·22-s − 2.13e7·23-s + 2.42e7·24-s + 6.90e7·25-s + 7.68e7·26-s − 1.43e7·27-s + 9.44e5·28-s + ⋯ |
| L(s) = 1 | + 0.886·2-s − 0.577·3-s − 0.213·4-s + 1.55·5-s − 0.511·6-s − 0.0485·7-s − 1.07·8-s + 0.333·9-s + 1.37·10-s − 1.04·11-s + 0.123·12-s + 1.43·13-s − 0.0430·14-s − 0.897·15-s − 0.740·16-s + 0.640·17-s + 0.295·18-s − 1.03·19-s − 0.332·20-s + 0.0280·21-s − 0.926·22-s − 0.692·23-s + 0.621·24-s + 1.41·25-s + 1.26·26-s − 0.192·27-s + 0.0103·28-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
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 40.1T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.08e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.15e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.58e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.91e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.75e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.12e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.13e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 6.39e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.05e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.08e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.74e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.55e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.53e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.48e8T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.86e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.74e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.81e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.43e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.94e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.81e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.07e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.86e10T + 7.15e21T^{2} \) |
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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
−10.24133105356703774476444400570, −9.354997709451856074590449886349, −8.220615928966522624852682370126, −6.42501292098782524920092353840, −5.83621794956066600265449695319, −5.16214224709590773071383205535, −3.92876253141981113412555710678, −2.62451177212707462708407617829, −1.42210196764692269943719539655, 0,
1.42210196764692269943719539655, 2.62451177212707462708407617829, 3.92876253141981113412555710678, 5.16214224709590773071383205535, 5.83621794956066600265449695319, 6.42501292098782524920092353840, 8.220615928966522624852682370126, 9.354997709451856074590449886349, 10.24133105356703774476444400570
