| L(s) = 1 | + 27.2·2-s − 243·3-s − 1.30e3·4-s − 8.75e3·5-s − 6.61e3·6-s − 6.82e4·7-s − 9.13e4·8-s + 5.90e4·9-s − 2.38e5·10-s + 6.64e5·11-s + 3.17e5·12-s − 1.65e6·13-s − 1.85e6·14-s + 2.12e6·15-s + 1.90e5·16-s + 1.11e7·17-s + 1.60e6·18-s − 8.29e5·19-s + 1.14e7·20-s + 1.65e7·21-s + 1.80e7·22-s + 1.30e6·23-s + 2.21e7·24-s + 2.77e7·25-s − 4.50e7·26-s − 1.43e7·27-s + 8.91e7·28-s + ⋯ |
| L(s) = 1 | + 0.601·2-s − 0.577·3-s − 0.638·4-s − 1.25·5-s − 0.347·6-s − 1.53·7-s − 0.985·8-s + 0.333·9-s − 0.753·10-s + 1.24·11-s + 0.368·12-s − 1.23·13-s − 0.922·14-s + 0.722·15-s + 0.0454·16-s + 1.90·17-s + 0.200·18-s − 0.0768·19-s + 0.799·20-s + 0.885·21-s + 0.747·22-s + 0.0422·23-s + 0.568·24-s + 0.568·25-s − 0.743·26-s − 0.192·27-s + 0.979·28-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 | \( 1 - 27.2T + 2.04e3T^{2} \) |
| 5 | \( 1 + 8.75e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.82e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 6.64e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.65e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.11e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.29e5T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.30e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 4.39e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.84e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.52e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.89e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.11e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.06e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.47e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 2.03e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.18e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.90e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.41e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.77e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.46e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.60e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.44e10T + 7.15e21T^{2} \) |
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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
−9.938101755376230997907519184390, −9.459168950387966417622699632209, −8.016280771268722345531168280998, −6.89708143654857606271358583594, −5.94146991010975934186396559243, −4.75224592371613339839789374911, −3.75721846454186107576946757452, −3.14680488394737555485484591828, −0.820125545011874894945188585694, 0,
0.820125545011874894945188585694, 3.14680488394737555485484591828, 3.75721846454186107576946757452, 4.75224592371613339839789374911, 5.94146991010975934186396559243, 6.89708143654857606271358583594, 8.016280771268722345531168280998, 9.459168950387966417622699632209, 9.938101755376230997907519184390
