| L(s) = 1 | + 2.78·2-s − 9·3-s − 24.2·4-s + 24.4·5-s − 25.0·6-s − 110.·7-s − 156.·8-s + 81·9-s + 67.9·10-s + 131.·11-s + 218.·12-s + 140.·13-s − 307.·14-s − 219.·15-s + 341.·16-s + 1.63e3·17-s + 225.·18-s + 262.·19-s − 592.·20-s + 996.·21-s + 365.·22-s + 590.·23-s + 1.40e3·24-s − 2.52e3·25-s + 390.·26-s − 729·27-s + 2.68e3·28-s + ⋯ |
| L(s) = 1 | + 0.491·2-s − 0.577·3-s − 0.758·4-s + 0.437·5-s − 0.283·6-s − 0.854·7-s − 0.864·8-s + 0.333·9-s + 0.214·10-s + 0.327·11-s + 0.437·12-s + 0.230·13-s − 0.419·14-s − 0.252·15-s + 0.333·16-s + 1.37·17-s + 0.163·18-s + 0.166·19-s − 0.331·20-s + 0.493·21-s + 0.161·22-s + 0.232·23-s + 0.499·24-s − 0.808·25-s + 0.113·26-s − 0.192·27-s + 0.647·28-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 | \( 1 - 2.78T + 32T^{2} \) |
| 5 | \( 1 - 24.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 110.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 131.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 140.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.63e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 262.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 590.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 990.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.28e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.63e4T + 8.58e9T^{2} \) |
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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
−9.821635368593833391179112396975, −9.103216707657089674940033682335, −7.957328059461559298981569999567, −6.64780801632413371117897739549, −5.87224965459814040532867102478, −5.12533663036164767220508560701, −3.94685660291359824966914155766, −3.04272748041030195007741694078, −1.22266088618573924231656703935, 0,
1.22266088618573924231656703935, 3.04272748041030195007741694078, 3.94685660291359824966914155766, 5.12533663036164767220508560701, 5.87224965459814040532867102478, 6.64780801632413371117897739549, 7.957328059461559298981569999567, 9.103216707657089674940033682335, 9.821635368593833391179112396975
