| L(s) = 1 | + 21.5·2-s + 337.·4-s − 125·5-s + 77.0·7-s + 4.51e3·8-s − 2.69e3·10-s − 3.71e3·11-s − 9.90e3·13-s + 1.66e3·14-s + 5.42e4·16-s − 1.22e4·17-s − 4.22e4·19-s − 4.21e4·20-s − 8.00e4·22-s − 8.37e4·23-s + 1.56e4·25-s − 2.13e5·26-s + 2.59e4·28-s − 7.79e4·29-s + 2.55e5·31-s + 5.93e5·32-s − 2.63e5·34-s − 9.62e3·35-s − 5.05e5·37-s − 9.10e5·38-s − 5.64e5·40-s + 3.62e5·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s − 0.447·5-s + 0.0848·7-s + 3.12·8-s − 0.852·10-s − 0.840·11-s − 1.25·13-s + 0.161·14-s + 3.31·16-s − 0.603·17-s − 1.41·19-s − 1.17·20-s − 1.60·22-s − 1.43·23-s + 0.199·25-s − 2.38·26-s + 0.223·28-s − 0.593·29-s + 1.54·31-s + 3.19·32-s − 1.15·34-s − 0.0379·35-s − 1.63·37-s − 2.69·38-s − 1.39·40-s + 0.822·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 21.5T + 128T^{2} \) |
| 7 | \( 1 - 77.0T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.71e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.90e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.22e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.22e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.79e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.55e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.21e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.72e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.67e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.44e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.99e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.05e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.21e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.16e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.63e7T + 8.07e13T^{2} \) |
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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
−10.11478929369887810660655059504, −8.325314202116247956158649614450, −7.40462983509168322691559358631, −6.55738026807345040095472240191, −5.55312810706333607316332737874, −4.63791481654522467518189293168, −3.98566740844764684286137237805, −2.70076621428952847809488336481, −2.02866195139391709377620896556, 0,
2.02866195139391709377620896556, 2.70076621428952847809488336481, 3.98566740844764684286137237805, 4.63791481654522467518189293168, 5.55312810706333607316332737874, 6.55738026807345040095472240191, 7.40462983509168322691559358631, 8.325314202116247956158649614450, 10.11478929369887810660655059504
