| L(s) = 1 | + 5-s + 0.561·7-s + 11-s + 5.12·13-s − 1.43·17-s − 6.56·19-s − 1.12·23-s + 25-s + 4.56·29-s + 3.68·31-s + 0.561·35-s − 10.8·37-s + 10·41-s + 3.12·43-s + 1.12·47-s − 6.68·49-s + 8.56·53-s + 55-s + 11.3·59-s + 0.561·61-s + 5.12·65-s − 6.56·71-s + 13.1·73-s + 0.561·77-s − 9.12·79-s + 10·83-s − 1.43·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.212·7-s + 0.301·11-s + 1.42·13-s − 0.348·17-s − 1.50·19-s − 0.234·23-s + 0.200·25-s + 0.847·29-s + 0.661·31-s + 0.0949·35-s − 1.77·37-s + 1.56·41-s + 0.476·43-s + 0.163·47-s − 0.954·49-s + 1.17·53-s + 0.134·55-s + 1.48·59-s + 0.0718·61-s + 0.635·65-s − 0.778·71-s + 1.53·73-s + 0.0639·77-s − 1.02·79-s + 1.09·83-s − 0.156·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.475463913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.475463913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 0.561T + 7T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.561T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 8.87T + 97T^{2} \) |
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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
−8.004317501951869163179482144000, −6.97866757285775060201608639404, −6.39541044971976803353240890655, −5.93402182500409184376574004642, −5.04136918871565751920425793419, −4.22142590215421450563611681585, −3.64851776719992005557794337299, −2.55394417840392065262493232807, −1.79589642509820439833599751390, −0.802907651325851862520515415274,
0.802907651325851862520515415274, 1.79589642509820439833599751390, 2.55394417840392065262493232807, 3.64851776719992005557794337299, 4.22142590215421450563611681585, 5.04136918871565751920425793419, 5.93402182500409184376574004642, 6.39541044971976803353240890655, 6.97866757285775060201608639404, 8.004317501951869163179482144000
