| L(s) = 1 | + 2.51·3-s + 2.85·5-s + 3.34·9-s − 2.34·11-s + 0.859·13-s + 7.20·15-s − 4.34·17-s + 19-s + 4.85·23-s + 3.17·25-s + 0.859·27-s + 9.37·29-s + 7.37·31-s − 5.89·33-s + 2.17·37-s + 2.16·39-s − 6.69·41-s + 0.353·43-s + 9.55·45-s − 5.54·47-s − 10.9·51-s + 9.55·53-s − 6.69·55-s + 2.51·57-s + 14.5·59-s + 12.9·61-s + 2.45·65-s + ⋯ |
| L(s) = 1 | + 1.45·3-s + 1.27·5-s + 1.11·9-s − 0.705·11-s + 0.238·13-s + 1.85·15-s − 1.05·17-s + 0.229·19-s + 1.01·23-s + 0.635·25-s + 0.165·27-s + 1.74·29-s + 1.32·31-s − 1.02·33-s + 0.357·37-s + 0.346·39-s − 1.04·41-s + 0.0539·43-s + 1.42·45-s − 0.808·47-s − 1.53·51-s + 1.31·53-s − 0.902·55-s + 0.333·57-s + 1.89·59-s + 1.65·61-s + 0.304·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.149689379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.149689379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.51T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 0.859T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 0.353T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 2.23T + 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 - 3.31T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.487990110309229378319677790607, −8.138329237402858362441296685510, −6.96574417639384905792170847121, −6.51032919360125665819937545831, −5.43759133939516394622532790395, −4.73520529111362876360908983046, −3.68092717547008909006322633128, −2.56478679926460993918607264808, −2.45626779252593197297231129562, −1.19378433966741929290472365821,
1.19378433966741929290472365821, 2.45626779252593197297231129562, 2.56478679926460993918607264808, 3.68092717547008909006322633128, 4.73520529111362876360908983046, 5.43759133939516394622532790395, 6.51032919360125665819937545831, 6.96574417639384905792170847121, 8.138329237402858362441296685510, 8.487990110309229378319677790607
