| L(s) = 1 | + 7.07·2-s + 9·3-s + 18.1·4-s − 24.1·5-s + 63.7·6-s + 54.4·7-s − 98.3·8-s + 81·9-s − 171.·10-s + 458.·11-s + 163.·12-s − 906.·13-s + 385.·14-s − 217.·15-s − 1.27e3·16-s − 1.56e3·17-s + 573.·18-s + 289.·19-s − 437.·20-s + 490.·21-s + 3.24e3·22-s + 1.43e3·23-s − 884.·24-s − 2.54e3·25-s − 6.41e3·26-s + 729·27-s + 986.·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.565·4-s − 0.432·5-s + 0.722·6-s + 0.420·7-s − 0.543·8-s + 0.333·9-s − 0.541·10-s + 1.14·11-s + 0.326·12-s − 1.48·13-s + 0.525·14-s − 0.249·15-s − 1.24·16-s − 1.31·17-s + 0.417·18-s + 0.183·19-s − 0.244·20-s + 0.242·21-s + 1.43·22-s + 0.564·23-s − 0.313·24-s − 0.812·25-s − 1.86·26-s + 0.192·27-s + 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 157 | \( 1 - 2.46e4T \) |
| good | 2 | \( 1 - 7.07T + 32T^{2} \) |
| 5 | \( 1 + 24.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 54.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 458.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 906.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 289.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 234.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.77e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.74e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.37e4T + 8.58e9T^{2} \) |
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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
−9.472322072117590061055963784202, −9.071318416206927586365792701842, −7.73830080380965225157820580450, −6.94541799712085141998301636871, −5.82560985561637189775701905034, −4.60479912016887482054314775027, −4.13822859319326527066845174493, −2.99194758531410592883272888799, −1.89509080353999540825610818664, 0,
1.89509080353999540825610818664, 2.99194758531410592883272888799, 4.13822859319326527066845174493, 4.60479912016887482054314775027, 5.82560985561637189775701905034, 6.94541799712085141998301636871, 7.73830080380965225157820580450, 9.071318416206927586365792701842, 9.472322072117590061055963784202
