| L(s) = 1 | − 32·2-s + 594.·3-s + 1.02e3·4-s − 490.·5-s − 1.90e4·6-s − 8.55e4·7-s − 3.27e4·8-s + 1.76e5·9-s + 1.57e4·10-s + 1.61e5·11-s + 6.08e5·12-s − 1.65e6·13-s + 2.73e6·14-s − 2.91e5·15-s + 1.04e6·16-s + 1.85e6·17-s − 5.63e6·18-s − 7.19e6·19-s − 5.02e5·20-s − 5.08e7·21-s − 5.15e6·22-s − 1.34e7·23-s − 1.94e7·24-s − 4.85e7·25-s + 5.29e7·26-s − 5.45e5·27-s − 8.76e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.0702·5-s − 0.998·6-s − 1.92·7-s − 0.353·8-s + 0.994·9-s + 0.0496·10-s + 0.301·11-s + 0.706·12-s − 1.23·13-s + 1.36·14-s − 0.0991·15-s + 0.250·16-s + 0.317·17-s − 0.703·18-s − 0.666·19-s − 0.0351·20-s − 2.71·21-s − 0.213·22-s − 0.436·23-s − 0.499·24-s − 0.995·25-s + 0.874·26-s − 0.00731·27-s − 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 32T \) |
| 11 | \( 1 - 1.61e5T \) |
| good | 3 | \( 1 - 594.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 490.T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.55e4T + 1.97e9T^{2} \) |
| 13 | \( 1 + 1.65e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.19e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.34e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.42e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.80e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.39e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.10e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.31e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.38e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.27e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.24e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 6.59e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.20e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.78e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.25e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.30e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.69e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.57e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.30e9T + 7.15e21T^{2} \) |
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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
−14.89250587235452555893846277429, −13.50618201875281721111303340669, −12.27317417394405592249489277876, −9.946828978692869538618584615644, −9.379322072353758846973054049839, −7.935926732915081538182757548796, −6.54892532967006381065130027003, −3.53026342478156154646989043642, −2.36531542062071533781819865937, 0,
2.36531542062071533781819865937, 3.53026342478156154646989043642, 6.54892532967006381065130027003, 7.935926732915081538182757548796, 9.379322072353758846973054049839, 9.946828978692869538618584615644, 12.27317417394405592249489277876, 13.50618201875281721111303340669, 14.89250587235452555893846277429
