| L(s) = 1 | − 5.77·2-s − 94.6·4-s + 218.·5-s + 606.·7-s + 1.28e3·8-s − 1.26e3·10-s + 3.21e3·11-s − 8.14e3·13-s − 3.49e3·14-s + 4.69e3·16-s + 3.21e3·17-s + 6.69e3·19-s − 2.06e4·20-s − 1.85e4·22-s + 2.03e4·23-s − 3.03e4·25-s + 4.70e4·26-s − 5.73e4·28-s + 1.32e5·29-s − 1.57e5·31-s − 1.91e5·32-s − 1.85e4·34-s + 1.32e5·35-s − 5.58e5·37-s − 3.86e4·38-s + 2.80e5·40-s − 6.99e4·41-s + ⋯ |
| L(s) = 1 | − 0.510·2-s − 0.739·4-s + 0.781·5-s + 0.667·7-s + 0.887·8-s − 0.398·10-s + 0.727·11-s − 1.02·13-s − 0.340·14-s + 0.286·16-s + 0.158·17-s + 0.223·19-s − 0.578·20-s − 0.371·22-s + 0.348·23-s − 0.388·25-s + 0.524·26-s − 0.493·28-s + 1.01·29-s − 0.947·31-s − 1.03·32-s − 0.0809·34-s + 0.522·35-s − 1.81·37-s − 0.114·38-s + 0.693·40-s − 0.158·41-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 | \( 1 + 5.77T + 128T^{2} \) |
| 5 | \( 1 - 218.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 606.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.21e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.69e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.57e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.99e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.72e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.42e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.83e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.90e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.85e6T + 4.42e13T^{2} \) |
| 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
−9.338381541049401570478643428333, −8.520468494403794712330227429653, −7.63380689746388474802256217702, −6.64721634268551523166826369496, −5.34506770714810612355246235926, −4.77485089140231581867002033222, −3.55784976955890142614761100440, −2.04241191369854217149298061872, −1.23073089235269448169459011264, 0,
1.23073089235269448169459011264, 2.04241191369854217149298061872, 3.55784976955890142614761100440, 4.77485089140231581867002033222, 5.34506770714810612355246235926, 6.64721634268551523166826369496, 7.63380689746388474802256217702, 8.520468494403794712330227429653, 9.338381541049401570478643428333
