L(s) = 1 | − 54.1·2-s + 243·3-s + 884.·4-s − 9.49e3·5-s − 1.31e4·6-s − 2.10e4·7-s + 6.29e4·8-s + 5.90e4·9-s + 5.14e5·10-s − 8.43e5·11-s + 2.14e5·12-s + 2.09e6·13-s + 1.14e6·14-s − 2.30e6·15-s − 5.22e6·16-s − 5.90e6·17-s − 3.19e6·18-s + 4.14e6·19-s − 8.40e6·20-s − 5.12e6·21-s + 4.56e7·22-s − 4.05e7·23-s + 1.53e7·24-s + 4.13e7·25-s − 1.13e8·26-s + 1.43e7·27-s − 1.86e7·28-s + ⋯ |
L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.431·4-s − 1.35·5-s − 0.690·6-s − 0.474·7-s + 0.679·8-s + 0.333·9-s + 1.62·10-s − 1.57·11-s + 0.249·12-s + 1.56·13-s + 0.567·14-s − 0.784·15-s − 1.24·16-s − 1.00·17-s − 0.398·18-s + 0.384·19-s − 0.587·20-s − 0.273·21-s + 1.89·22-s − 1.31·23-s + 0.392·24-s + 0.847·25-s − 1.87·26-s + 0.192·27-s − 0.204·28-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 | \( 1 + 54.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 9.49e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.10e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.43e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.09e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.14e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.05e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.14e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.22e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.50e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.58e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.90e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.37e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.90e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 7.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.55e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.49e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 4.31e8T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.22e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.44e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.02e11T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.09e11T + 7.15e21T^{2} \) |
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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
−10.12393235259357006448592871963, −8.912041170078662463340774567607, −8.110650881281960858503073423069, −7.76355043030552786649198460456, −6.43814303466903670187498453144, −4.61152139179564486019194197840, −3.63023537485281250017744765250, −2.39541744424349480309844862935, −0.877397295172371023603587502067, 0,
0.877397295172371023603587502067, 2.39541744424349480309844862935, 3.63023537485281250017744765250, 4.61152139179564486019194197840, 6.43814303466903670187498453144, 7.76355043030552786649198460456, 8.110650881281960858503073423069, 8.912041170078662463340774567607, 10.12393235259357006448592871963