| L(s) = 1 | + 4.55·2-s − 107.·4-s + 540.·5-s − 1.23e3·7-s − 1.07e3·8-s + 2.46e3·10-s − 3.52e3·11-s − 4.83e3·13-s − 5.64e3·14-s + 8.84e3·16-s − 2.24e3·17-s + 5.31e4·19-s − 5.79e4·20-s − 1.60e4·22-s + 5.63e4·23-s + 2.13e5·25-s − 2.20e4·26-s + 1.32e5·28-s + 1.37e5·29-s − 1.21e5·31-s + 1.77e5·32-s − 1.02e4·34-s − 6.69e5·35-s − 4.76e5·37-s + 2.42e5·38-s − 5.79e5·40-s − 1.40e5·41-s + ⋯ |
| L(s) = 1 | + 0.402·2-s − 0.837·4-s + 1.93·5-s − 1.36·7-s − 0.740·8-s + 0.778·10-s − 0.797·11-s − 0.610·13-s − 0.549·14-s + 0.539·16-s − 0.110·17-s + 1.77·19-s − 1.61·20-s − 0.321·22-s + 0.966·23-s + 2.73·25-s − 0.245·26-s + 1.14·28-s + 1.04·29-s − 0.732·31-s + 0.957·32-s − 0.0446·34-s − 2.63·35-s − 1.54·37-s + 0.715·38-s − 1.43·40-s − 0.318·41-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 | \( 1 - 4.55T + 128T^{2} \) |
| 5 | \( 1 - 540.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.83e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.24e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.37e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.21e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.05e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.36e6T + 8.07e13T^{2} \) |
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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
−9.549100172532873763658266987644, −8.751597190249738078452875128750, −7.20841468349879057435239413947, −6.27671321886559272664357804160, −5.42996112346139640538553159259, −4.95663741990869793164724105474, −3.23663197268996183100376602014, −2.70315343055361772063440625363, −1.23274980586705725350414018262, 0,
1.23274980586705725350414018262, 2.70315343055361772063440625363, 3.23663197268996183100376602014, 4.95663741990869793164724105474, 5.42996112346139640538553159259, 6.27671321886559272664357804160, 7.20841468349879057435239413947, 8.751597190249738078452875128750, 9.549100172532873763658266987644
