| L(s) = 1 | − 1.56·2-s + 0.438·4-s + 5-s − 0.561·7-s + 2.43·8-s − 1.56·10-s + 1.43·11-s + 0.876·14-s − 4.68·16-s − 4.56·17-s − 7.12·19-s + 0.438·20-s − 2.24·22-s + 1.43·23-s + 25-s − 0.246·28-s + 8.24·29-s − 6·31-s + 2.43·32-s + 7.12·34-s − 0.561·35-s − 1.43·37-s + 11.1·38-s + 2.43·40-s + 4.56·41-s + 1.12·43-s + 0.630·44-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.219·4-s + 0.447·5-s − 0.212·7-s + 0.862·8-s − 0.493·10-s + 0.433·11-s + 0.234·14-s − 1.17·16-s − 1.10·17-s − 1.63·19-s + 0.0980·20-s − 0.478·22-s + 0.299·23-s + 0.200·25-s − 0.0465·28-s + 1.53·29-s − 1.07·31-s + 0.431·32-s + 1.22·34-s − 0.0949·35-s − 0.236·37-s + 1.80·38-s + 0.385·40-s + 0.712·41-s + 0.171·43-s + 0.0950·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7829549605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7829549605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.132440212592810593275888644316, −7.25205233852613308409661805611, −6.53151885810136376711864138728, −6.16741889807198677992386620483, −4.85762674651273764162901638050, −4.49649604719643185178212294231, −3.47583711628516405329842142870, −2.31190517990692117373700324639, −1.68939782714197712377292569486, −0.52657234839112931251656539288,
0.52657234839112931251656539288, 1.68939782714197712377292569486, 2.31190517990692117373700324639, 3.47583711628516405329842142870, 4.49649604719643185178212294231, 4.85762674651273764162901638050, 6.16741889807198677992386620483, 6.53151885810136376711864138728, 7.25205233852613308409661805611, 8.132440212592810593275888644316
