| L(s) = 1 | − 19.6·2-s + 257.·4-s − 206.·5-s + 1.44e3·7-s − 2.54e3·8-s + 4.05e3·10-s − 3.13e3·11-s + 1.87e3·13-s − 2.83e4·14-s + 1.70e4·16-s + 6.04e3·17-s + 4.19e4·19-s − 5.32e4·20-s + 6.16e4·22-s + 3.36e4·23-s − 3.54e4·25-s − 3.68e4·26-s + 3.71e5·28-s − 1.56e5·29-s + 1.19e5·31-s − 8.44e3·32-s − 1.18e5·34-s − 2.98e5·35-s − 2.77e5·37-s − 8.23e5·38-s + 5.26e5·40-s + 1.05e4·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s − 0.739·5-s + 1.58·7-s − 1.75·8-s + 1.28·10-s − 0.711·11-s + 0.237·13-s − 2.75·14-s + 1.03·16-s + 0.298·17-s + 1.40·19-s − 1.48·20-s + 1.23·22-s + 0.576·23-s − 0.453·25-s − 0.411·26-s + 3.20·28-s − 1.19·29-s + 0.720·31-s − 0.0455·32-s − 0.518·34-s − 1.17·35-s − 0.900·37-s − 2.43·38-s + 1.30·40-s + 0.0239·41-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 | \( 1 + 19.6T + 128T^{2} \) |
| 5 | \( 1 + 206.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.44e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.13e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.87e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.04e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.36e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.56e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.05e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.26e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.33e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.77e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.59e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.14e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.77e4T + 8.07e13T^{2} \) |
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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
−9.090528443529275399805823682891, −8.317820972798883930046511371242, −7.64535109882935260315694321113, −7.29152784119493033979230006537, −5.69838715160819085564423631062, −4.64140220252167729974903451906, −3.15200113588668956500338610216, −1.85556493586915829396587501130, −1.08587826525419208843876158710, 0,
1.08587826525419208843876158710, 1.85556493586915829396587501130, 3.15200113588668956500338610216, 4.64140220252167729974903451906, 5.69838715160819085564423631062, 7.29152784119493033979230006537, 7.64535109882935260315694321113, 8.317820972798883930046511371242, 9.090528443529275399805823682891
