| L(s) = 1 | + 4.32·2-s + 10.7·4-s + 5·5-s − 6.68·7-s + 11.8·8-s + 21.6·10-s + 11·11-s + 59.6·13-s − 28.9·14-s − 34.5·16-s + 131.·17-s + 52.1·19-s + 53.7·20-s + 47.6·22-s − 15.3·23-s + 25·25-s + 258.·26-s − 71.8·28-s + 59.6·29-s + 229.·31-s − 244.·32-s + 569.·34-s − 33.4·35-s − 241.·37-s + 225.·38-s + 59.3·40-s + 196.·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.34·4-s + 0.447·5-s − 0.360·7-s + 0.524·8-s + 0.684·10-s + 0.301·11-s + 1.27·13-s − 0.552·14-s − 0.539·16-s + 1.87·17-s + 0.629·19-s + 0.600·20-s + 0.461·22-s − 0.139·23-s + 0.200·25-s + 1.94·26-s − 0.484·28-s + 0.382·29-s + 1.32·31-s − 1.35·32-s + 2.87·34-s − 0.161·35-s − 1.07·37-s + 0.962·38-s + 0.234·40-s + 0.749·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.269621099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.269621099\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.32T + 8T^{2} \) |
| 7 | \( 1 + 6.68T + 343T^{2} \) |
| 13 | \( 1 - 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 59.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 195.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 517.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 763.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 269.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 895.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 855.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 391.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 829.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 412.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−10.71791092179487365379840913958, −9.807214377160078191333204561975, −8.765212219810799428291640706725, −7.53386075504684776083174707759, −6.32110383550839812029051493659, −5.85137751145547955940882992079, −4.84354051232063507586080016142, −3.65041079932563233381628322907, −2.96459595392755470193797719679, −1.30324663555597406655013391565,
1.30324663555597406655013391565, 2.96459595392755470193797719679, 3.65041079932563233381628322907, 4.84354051232063507586080016142, 5.85137751145547955940882992079, 6.32110383550839812029051493659, 7.53386075504684776083174707759, 8.765212219810799428291640706725, 9.807214377160078191333204561975, 10.71791092179487365379840913958
