| L(s) = 1 | − 0.425·2-s − 1.37·3-s − 1.81·4-s − 0.119·5-s + 0.584·6-s − 3.28·7-s + 1.62·8-s − 1.11·9-s + 0.0508·10-s + 11-s + 2.49·12-s + 3.12·13-s + 1.39·14-s + 0.163·15-s + 2.94·16-s + 17-s + 0.476·18-s + 3.73·19-s + 0.216·20-s + 4.50·21-s − 0.425·22-s + 9.46·23-s − 2.23·24-s − 4.98·25-s − 1.33·26-s + 5.64·27-s + 5.96·28-s + ⋯ |
| L(s) = 1 | − 0.301·2-s − 0.792·3-s − 0.909·4-s − 0.0533·5-s + 0.238·6-s − 1.23·7-s + 0.575·8-s − 0.372·9-s + 0.0160·10-s + 0.301·11-s + 0.720·12-s + 0.867·13-s + 0.373·14-s + 0.0422·15-s + 0.736·16-s + 0.242·17-s + 0.112·18-s + 0.857·19-s + 0.0484·20-s + 0.982·21-s − 0.0908·22-s + 1.97·23-s − 0.455·24-s − 0.997·25-s − 0.261·26-s + 1.08·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6827008105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6827008105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.425T + 2T^{2} \) |
| 3 | \( 1 + 1.37T + 3T^{2} \) |
| 5 | \( 1 + 0.119T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 3.11T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 8.97T + 97T^{2} \) |
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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
−7.80601113092381438169887504129, −7.13629058311567530115280262898, −6.28698976777962491923458231386, −5.79991369079355510936634132300, −5.16467853933406600594679054965, −4.33209164865023032238149313957, −3.46520608857083429664680969706, −2.95330179640996663809943546021, −1.29584275218500611756979290573, −0.50991092333639262770974482686,
0.50991092333639262770974482686, 1.29584275218500611756979290573, 2.95330179640996663809943546021, 3.46520608857083429664680969706, 4.33209164865023032238149313957, 5.16467853933406600594679054965, 5.79991369079355510936634132300, 6.28698976777962491923458231386, 7.13629058311567530115280262898, 7.80601113092381438169887504129
