L(s) = 1 | + 79.9·2-s + 243·3-s + 4.34e3·4-s + 3.57e3·5-s + 1.94e4·6-s + 3.42e4·7-s + 1.84e5·8-s + 5.90e4·9-s + 2.85e5·10-s + 5.61e5·11-s + 1.05e6·12-s − 8.41e5·13-s + 2.74e6·14-s + 8.68e5·15-s + 5.81e6·16-s − 1.34e6·17-s + 4.72e6·18-s − 1.30e6·19-s + 1.55e7·20-s + 8.33e6·21-s + 4.49e7·22-s + 5.35e7·23-s + 4.47e7·24-s − 3.60e7·25-s − 6.73e7·26-s + 1.43e7·27-s + 1.49e8·28-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 0.577·3-s + 2.12·4-s + 0.511·5-s + 1.02·6-s + 0.771·7-s + 1.98·8-s + 0.333·9-s + 0.903·10-s + 1.05·11-s + 1.22·12-s − 0.628·13-s + 1.36·14-s + 0.295·15-s + 1.38·16-s − 0.229·17-s + 0.589·18-s − 0.121·19-s + 1.08·20-s + 0.445·21-s + 1.85·22-s + 1.73·23-s + 1.14·24-s − 0.738·25-s − 1.11·26-s + 0.192·27-s + 1.63·28-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
\(L(6)\) |
\(\approx\) |
\(11.88143315\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.88143315\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 - 79.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.57e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.42e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.61e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 8.41e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.34e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.30e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.35e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.22e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.64e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.58e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.88e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.19e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.16e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.57e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 7.71e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.30e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.59e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.26e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.07e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.23e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.52e10T + 7.15e21T^{2} \) |
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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
−11.09004492454401455748984385994, −9.787949785050480369159699889081, −8.550629241256922648866226361678, −7.18655241877490319504781700523, −6.37853118979307573678767138255, −5.13437051301394414077894507609, −4.42794151383462559496605975247, −3.29714342335399070872678460636, −2.29017387670643206785266718984, −1.34193984115932125397687450791,
1.34193984115932125397687450791, 2.29017387670643206785266718984, 3.29714342335399070872678460636, 4.42794151383462559496605975247, 5.13437051301394414077894507609, 6.37853118979307573678767138255, 7.18655241877490319504781700523, 8.550629241256922648866226361678, 9.787949785050480369159699889081, 11.09004492454401455748984385994