| L(s) = 1 | − 685.·2-s + 2.20e3·3-s + 3.38e5·4-s + 7.77e5·5-s − 1.51e6·6-s − 1.15e7·7-s − 1.42e8·8-s − 1.24e8·9-s − 5.32e8·10-s + 7.73e8·11-s + 7.47e8·12-s + 3.88e9·13-s + 7.89e9·14-s + 1.71e9·15-s + 5.30e10·16-s − 1.45e10·17-s + 8.51e10·18-s − 3.40e10·19-s + 2.63e11·20-s − 2.54e10·21-s − 5.30e11·22-s + 5.08e10·23-s − 3.13e11·24-s − 1.59e11·25-s − 2.66e12·26-s − 5.59e11·27-s − 3.90e12·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.194·3-s + 2.58·4-s + 0.889·5-s − 0.367·6-s − 0.755·7-s − 2.99·8-s − 0.962·9-s − 1.68·10-s + 1.08·11-s + 0.501·12-s + 1.32·13-s + 1.43·14-s + 0.172·15-s + 3.08·16-s − 0.504·17-s + 1.82·18-s − 0.459·19-s + 2.29·20-s − 0.146·21-s − 2.06·22-s + 0.135·23-s − 0.581·24-s − 0.208·25-s − 2.50·26-s − 0.381·27-s − 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + 1.46e14T \) |
| good | 2 | \( 1 + 685.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 2.20e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 7.77e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.15e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 7.73e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.88e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.45e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.40e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.08e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 6.27e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 4.71e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.78e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.52e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.22e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 8.00e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.27e14T + 2.05e29T^{2} \) |
| 61 | \( 1 + 6.49e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.88e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 2.17e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.62e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 4.02e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.74e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.55e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.41e17T + 5.95e33T^{2} \) |
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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
−10.82299738782308429290340806302, −9.591684530192325060388474936953, −9.062769323784152989201481653085, −8.124208642879462957301353916048, −6.47792782810714361669224741480, −6.12629220032150570566939813984, −3.36052626764667355982930602636, −2.17144477269384339490516125602, −1.18908440393633179850925768884, 0,
1.18908440393633179850925768884, 2.17144477269384339490516125602, 3.36052626764667355982930602636, 6.12629220032150570566939813984, 6.47792782810714361669224741480, 8.124208642879462957301353916048, 9.062769323784152989201481653085, 9.591684530192325060388474936953, 10.82299738782308429290340806302
