| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 7.25·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 14.5·10-s + 6.99·11-s − 12·12-s − 6.86·13-s + 14·14-s − 21.7·15-s + 16·16-s + 43.1·17-s + 18·18-s − 19·19-s + 29.0·20-s − 21·21-s + 13.9·22-s + 198.·23-s − 24·24-s − 72.3·25-s − 13.7·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.649·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.459·10-s + 0.191·11-s − 0.288·12-s − 0.146·13-s + 0.267·14-s − 0.374·15-s + 0.250·16-s + 0.615·17-s + 0.235·18-s − 0.229·19-s + 0.324·20-s − 0.218·21-s + 0.135·22-s + 1.80·23-s − 0.204·24-s − 0.578·25-s − 0.103·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.409607051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.409607051\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 7.25T + 125T^{2} \) |
| 11 | \( 1 - 6.99T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.86T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 29.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 39.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 94.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 31.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 280.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 975.T + 9.12e5T^{2} \) |
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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
−10.02159863679279011689978661647, −9.194525498489806474381475814585, −7.997393523414776526566671573691, −7.03533062747303681813241794800, −6.21581599676082251273819167046, −5.37134943086596578576044386797, −4.69295911333345613633247435834, −3.48039673816754829414151966094, −2.20965914515537405905342072335, −1.01378992586530547788583438467,
1.01378992586530547788583438467, 2.20965914515537405905342072335, 3.48039673816754829414151966094, 4.69295911333345613633247435834, 5.37134943086596578576044386797, 6.21581599676082251273819167046, 7.03533062747303681813241794800, 7.997393523414776526566671573691, 9.194525498489806474381475814585, 10.02159863679279011689978661647
