| L(s) = 1 | + 3.64·5-s + 3.17·7-s + 0.962·11-s + 5.85·13-s + 2.68·17-s − 5.50·19-s − 23-s + 8.29·25-s − 1.85·29-s − 1.85·31-s + 11.5·35-s − 6.67·37-s − 9.14·41-s + 7.57·43-s − 8.34·47-s + 3.06·49-s − 5.99·53-s + 3.50·55-s − 8.80·59-s + 1.03·61-s + 21.3·65-s − 1.08·67-s − 13.1·71-s + 13.2·73-s + 3.05·77-s − 4.53·79-s − 3.03·83-s + ⋯ |
| L(s) = 1 | + 1.63·5-s + 1.19·7-s + 0.290·11-s + 1.62·13-s + 0.650·17-s − 1.26·19-s − 0.208·23-s + 1.65·25-s − 0.344·29-s − 0.333·31-s + 1.95·35-s − 1.09·37-s − 1.42·41-s + 1.15·43-s − 1.21·47-s + 0.438·49-s − 0.823·53-s + 0.473·55-s − 1.14·59-s + 0.132·61-s + 2.64·65-s − 0.132·67-s − 1.56·71-s + 1.54·73-s + 0.348·77-s − 0.510·79-s − 0.333·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850743566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850743566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + 6.67T + 37T^{2} \) |
| 41 | \( 1 + 9.14T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 1.08T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.273126892047145659342332319555, −8.647296970941885024653077660278, −7.991195487004334696541323024746, −6.74078374463219385787527893985, −6.05959431853814355327209549084, −5.42261032142927870528859413032, −4.50166017300869695527825766997, −3.34668603118406206643199305416, −1.91184610810155729184832578275, −1.45983876810372015584272855720,
1.45983876810372015584272855720, 1.91184610810155729184832578275, 3.34668603118406206643199305416, 4.50166017300869695527825766997, 5.42261032142927870528859413032, 6.05959431853814355327209549084, 6.74078374463219385787527893985, 7.991195487004334696541323024746, 8.647296970941885024653077660278, 9.273126892047145659342332319555
