| L(s) = 1 | + 7.89·2-s − 243·3-s − 1.98e3·4-s − 1.40e3·5-s − 1.91e3·6-s + 2.51e4·7-s − 3.18e4·8-s + 5.90e4·9-s − 1.10e4·10-s + 2.51e5·11-s + 4.82e5·12-s + 1.97e6·13-s + 1.98e5·14-s + 3.41e5·15-s + 3.81e6·16-s + 3.84e5·17-s + 4.65e5·18-s − 1.00e7·19-s + 2.79e6·20-s − 6.10e6·21-s + 1.98e6·22-s + 5.12e7·23-s + 7.73e6·24-s − 4.68e7·25-s + 1.56e7·26-s − 1.43e7·27-s − 4.98e7·28-s + ⋯ |
| L(s) = 1 | + 0.174·2-s − 0.577·3-s − 0.969·4-s − 0.201·5-s − 0.100·6-s + 0.564·7-s − 0.343·8-s + 0.333·9-s − 0.0350·10-s + 0.471·11-s + 0.559·12-s + 1.47·13-s + 0.0984·14-s + 0.116·15-s + 0.909·16-s + 0.0656·17-s + 0.0581·18-s − 0.926·19-s + 0.195·20-s − 0.325·21-s + 0.0822·22-s + 1.66·23-s + 0.198·24-s − 0.959·25-s + 0.257·26-s − 0.192·27-s − 0.547·28-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
| \(L(6)\) |
\(\approx\) |
\(1.495863281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495863281\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 7.89T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.40e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.51e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.51e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.97e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.84e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.00e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.12e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.79e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.54e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.49e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.77e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.32e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.18e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 8.62e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.42e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 8.96e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.87e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.71e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.86e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.87e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.39e10T + 7.15e21T^{2} \) |
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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
−10.96738070525983641123032325123, −9.543036365991338920390466729427, −8.733836151468912708654703392358, −7.72432736237899597422301109897, −6.31250150252556069735030780863, −5.40450830947861044836662338004, −4.32730561400279496953517257077, −3.54654293494057771689938942282, −1.61054320209426454993316060372, −0.59623679698766838268561880254,
0.59623679698766838268561880254, 1.61054320209426454993316060372, 3.54654293494057771689938942282, 4.32730561400279496953517257077, 5.40450830947861044836662338004, 6.31250150252556069735030780863, 7.72432736237899597422301109897, 8.733836151468912708654703392358, 9.543036365991338920390466729427, 10.96738070525983641123032325123
