| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 1.13e4·5-s + 1.68e4·7-s − 3.27e4·8-s + 3.64e5·10-s + 1.31e5·11-s − 1.32e6·13-s − 5.37e5·14-s + 1.04e6·16-s − 5.49e6·17-s + 2.05e7·19-s − 1.16e7·20-s − 4.20e6·22-s − 4.21e6·23-s + 8.07e7·25-s + 4.23e7·26-s + 1.72e7·28-s + 4.69e7·29-s + 8.70e7·31-s − 3.35e7·32-s + 1.75e8·34-s − 1.91e8·35-s − 4.82e7·37-s − 6.56e8·38-s + 3.73e8·40-s − 2.74e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.62·5-s + 0.377·7-s − 0.353·8-s + 1.15·10-s + 0.245·11-s − 0.988·13-s − 0.267·14-s + 0.250·16-s − 0.938·17-s + 1.90·19-s − 0.814·20-s − 0.173·22-s − 0.136·23-s + 1.65·25-s + 0.699·26-s + 0.188·28-s + 0.425·29-s + 0.546·31-s − 0.176·32-s + 0.663·34-s − 0.615·35-s − 0.114·37-s − 1.34·38-s + 0.576·40-s − 0.370·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 + 1.13e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.31e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.32e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.49e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.05e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.21e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.69e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 8.70e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.82e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.74e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.07e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.70e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.40e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.62e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.61e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.51e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.63e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.40e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.25e8T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.67e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.23e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 6.53e10T + 7.15e21T^{2} \) |
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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
−10.84020077682868810291346164874, −9.597826930768406237118519808730, −8.479452773631601398590456360200, −7.62064269521321419585306641539, −6.91102594187186042085266673120, −5.08961598348188886580710303337, −3.91985540995023405485982963450, −2.66488790348156816293170995876, −1.02737258875988037046435226312, 0,
1.02737258875988037046435226312, 2.66488790348156816293170995876, 3.91985540995023405485982963450, 5.08961598348188886580710303337, 6.91102594187186042085266673120, 7.62064269521321419585306641539, 8.479452773631601398590456360200, 9.597826930768406237118519808730, 10.84020077682868810291346164874
