| L(s) = 1 | − 7.43·2-s − 72.6·4-s − 171.·5-s + 1.52e3·7-s + 1.49e3·8-s + 1.27e3·10-s + 1.63e3·11-s − 7.31e3·13-s − 1.13e4·14-s − 1.79e3·16-s − 3.32e4·17-s + 5.55e4·19-s + 1.24e4·20-s − 1.21e4·22-s + 5.18e4·23-s − 4.86e4·25-s + 5.44e4·26-s − 1.11e5·28-s − 1.03e4·29-s + 9.27e4·31-s − 1.77e5·32-s + 2.47e5·34-s − 2.62e5·35-s − 5.04e5·37-s − 4.13e5·38-s − 2.56e5·40-s − 6.35e5·41-s + ⋯ |
| L(s) = 1 | − 0.657·2-s − 0.567·4-s − 0.614·5-s + 1.68·7-s + 1.03·8-s + 0.404·10-s + 0.371·11-s − 0.923·13-s − 1.10·14-s − 0.109·16-s − 1.63·17-s + 1.85·19-s + 0.349·20-s − 0.244·22-s + 0.889·23-s − 0.622·25-s + 0.607·26-s − 0.957·28-s − 0.0790·29-s + 0.559·31-s − 0.958·32-s + 1.07·34-s − 1.03·35-s − 1.63·37-s − 1.22·38-s − 0.633·40-s − 1.44·41-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 7.43T + 128T^{2} \) |
| 5 | \( 1 + 171.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.52e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.31e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.03e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.27e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.04e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.35e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.24e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.67e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 7.27e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.81e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 8.50e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.16e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.67e7T + 8.07e13T^{2} \) |
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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
−9.120501442307573438867568660834, −8.386664824065957945069862895710, −7.68122768246290152420127537255, −6.96950629957387586736696412547, −5.05764309485143479421947836758, −4.83041318403485266090124737046, −3.66183699015351955749325377384, −2.03395429832629460110561489728, −1.09329065539619481207494013632, 0,
1.09329065539619481207494013632, 2.03395429832629460110561489728, 3.66183699015351955749325377384, 4.83041318403485266090124737046, 5.05764309485143479421947836758, 6.96950629957387586736696412547, 7.68122768246290152420127537255, 8.386664824065957945069862895710, 9.120501442307573438867568660834
