| L(s) = 1 | + 17.9·2-s + 194.·4-s + 475.·5-s + 943.·7-s + 1.20e3·8-s + 8.53e3·10-s + 645.·11-s − 3.51e3·13-s + 1.69e4·14-s − 3.37e3·16-s − 1.66e4·17-s + 1.08e4·19-s + 9.25e4·20-s + 1.15e4·22-s + 3.34e4·23-s + 1.47e5·25-s − 6.31e4·26-s + 1.83e5·28-s − 1.02e4·29-s + 2.95e5·31-s − 2.14e5·32-s − 2.99e5·34-s + 4.48e5·35-s + 5.69e5·37-s + 1.95e5·38-s + 5.70e5·40-s + 2.64e5·41-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.52·4-s + 1.69·5-s + 1.03·7-s + 0.828·8-s + 2.69·10-s + 0.146·11-s − 0.443·13-s + 1.65·14-s − 0.205·16-s − 0.823·17-s + 0.364·19-s + 2.58·20-s + 0.232·22-s + 0.572·23-s + 1.88·25-s − 0.704·26-s + 1.58·28-s − 0.0781·29-s + 1.78·31-s − 1.15·32-s − 1.30·34-s + 1.76·35-s + 1.84·37-s + 0.578·38-s + 1.40·40-s + 0.599·41-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
| \(L(4)\) |
\(\approx\) |
\(10.15011288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.15011288\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 17.9T + 128T^{2} \) |
| 5 | \( 1 - 475.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 943.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 645.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.51e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.08e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.02e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.95e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.69e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.64e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.91e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.10e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.86e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.14e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.70e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.96e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.54e7T + 8.07e13T^{2} \) |
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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
−9.764033692031375599337660628115, −8.976049701469473126151182549695, −7.63449495459514966045515143181, −6.42439618908970338834900677545, −5.93931857290590477547305665187, −4.90216189446394596878721722456, −4.49461640329618583024582789416, −2.85452172607169874763195698940, −2.22162405186141284600640629170, −1.18840170127193313791929897332,
1.18840170127193313791929897332, 2.22162405186141284600640629170, 2.85452172607169874763195698940, 4.49461640329618583024582789416, 4.90216189446394596878721722456, 5.93931857290590477547305665187, 6.42439618908970338834900677545, 7.63449495459514966045515143181, 8.976049701469473126151182549695, 9.764033692031375599337660628115
