| L(s) = 1 | + 10.4·2-s − 18.9·4-s + 410.·5-s + 1.24e3·7-s − 1.53e3·8-s + 4.28e3·10-s + 2.42e3·11-s + 1.13e4·13-s + 1.30e4·14-s − 1.35e4·16-s + 1.86e4·17-s + 9.29e3·19-s − 7.77e3·20-s + 2.52e4·22-s + 1.02e5·23-s + 9.01e4·25-s + 1.18e5·26-s − 2.36e4·28-s − 1.84e5·29-s − 2.57e5·31-s + 5.44e4·32-s + 1.94e5·34-s + 5.12e5·35-s − 9.60e4·37-s + 9.70e4·38-s − 6.29e5·40-s + 6.77e5·41-s + ⋯ |
| L(s) = 1 | + 0.923·2-s − 0.148·4-s + 1.46·5-s + 1.37·7-s − 1.05·8-s + 1.35·10-s + 0.548·11-s + 1.42·13-s + 1.27·14-s − 0.830·16-s + 0.919·17-s + 0.310·19-s − 0.217·20-s + 0.506·22-s + 1.75·23-s + 1.15·25-s + 1.31·26-s − 0.203·28-s − 1.40·29-s − 1.55·31-s + 0.293·32-s + 0.849·34-s + 2.02·35-s − 0.311·37-s + 0.287·38-s − 1.55·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.441276841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.441276841\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 10.4T + 128T^{2} \) |
| 5 | \( 1 - 410.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.24e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.13e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.29e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.02e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.84e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.57e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.60e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.60e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.94e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.42e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 8.95e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.17e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.73e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.01e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.34e7T + 8.07e13T^{2} \) |
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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
−9.397925304696734952120555061375, −9.108671320411667351899277553468, −7.960425773709288701132816574042, −6.65046951553108188263823163636, −5.53646867736550526900155021772, −5.39311651584854090339321904994, −4.14916395666373727149043485856, −3.14183932018275245184439253411, −1.77473746194373316584244821703, −1.08642241067204049565123258280,
1.08642241067204049565123258280, 1.77473746194373316584244821703, 3.14183932018275245184439253411, 4.14916395666373727149043485856, 5.39311651584854090339321904994, 5.53646867736550526900155021772, 6.65046951553108188263823163636, 7.960425773709288701132816574042, 9.108671320411667351899277553468, 9.397925304696734952120555061375
