| L(s) = 1 | + 3.24·2-s − 117.·4-s − 168.·5-s + 149.·7-s − 796.·8-s − 545.·10-s + 18.5·11-s + 1.02e4·13-s + 486.·14-s + 1.24e4·16-s + 2.59e4·17-s − 1.61e4·19-s + 1.97e4·20-s + 60.1·22-s − 1.21e4·23-s − 4.98e4·25-s + 3.31e4·26-s − 1.75e4·28-s + 9.28e4·29-s − 5.22e4·31-s + 1.42e5·32-s + 8.41e4·34-s − 2.51e4·35-s − 7.66e4·37-s − 5.24e4·38-s + 1.33e5·40-s − 1.08e5·41-s + ⋯ |
| L(s) = 1 | + 0.286·2-s − 0.917·4-s − 0.601·5-s + 0.165·7-s − 0.550·8-s − 0.172·10-s + 0.00420·11-s + 1.28·13-s + 0.0473·14-s + 0.759·16-s + 1.27·17-s − 0.540·19-s + 0.551·20-s + 0.00120·22-s − 0.208·23-s − 0.638·25-s + 0.369·26-s − 0.151·28-s + 0.707·29-s − 0.314·31-s + 0.768·32-s + 0.367·34-s − 0.0992·35-s − 0.248·37-s − 0.155·38-s + 0.330·40-s − 0.246·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 3.24T + 128T^{2} \) |
| 5 | \( 1 + 168.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 149.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 18.5T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.61e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 9.28e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.22e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.66e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.08e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.91e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.33e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.47e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.71e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.06e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.83e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.07e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.23e6T + 8.07e13T^{2} \) |
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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
−10.64773616549715373558045243083, −9.604954041479929333433712003233, −8.492833585946875139478000599524, −7.88238982733933290580205888346, −6.32182602270299970830295828776, −5.25813660080119140097132036201, −4.08836457573597893513387823150, −3.30122256094787169411787415439, −1.30060382620511014207829667033, 0,
1.30060382620511014207829667033, 3.30122256094787169411787415439, 4.08836457573597893513387823150, 5.25813660080119140097132036201, 6.32182602270299970830295828776, 7.88238982733933290580205888346, 8.492833585946875139478000599524, 9.604954041479929333433712003233, 10.64773616549715373558045243083
