| L(s) = 1 | − 5·5-s + 12.8·7-s + 49.7·11-s − 52.6·13-s − 84.6·17-s + 26.6·19-s + 136.·23-s + 25·25-s − 6·29-s − 47.1·31-s − 64.3·35-s + 344.·37-s + 43.2·41-s − 252·43-s + 306.·47-s − 177.·49-s + 455.·53-s − 248.·55-s + 708.·59-s − 652.·61-s + 263.·65-s − 704.·67-s − 531.·71-s + 57.6·73-s + 640·77-s − 429.·79-s + 227.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.694·7-s + 1.36·11-s − 1.12·13-s − 1.20·17-s + 0.321·19-s + 1.23·23-s + 0.200·25-s − 0.0384·29-s − 0.273·31-s − 0.310·35-s + 1.52·37-s + 0.164·41-s − 0.893·43-s + 0.949·47-s − 0.517·49-s + 1.17·53-s − 0.609·55-s + 1.56·59-s − 1.36·61-s + 0.501·65-s − 1.28·67-s − 0.888·71-s + 0.0923·73-s + 0.947·77-s − 0.611·79-s + 0.301·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.114085951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114085951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 12.8T + 343T^{2} \) |
| 11 | \( 1 - 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 43.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252T + 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 708.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 652.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 531.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 57.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 429.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 227.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 152.T + 9.12e5T^{2} \) |
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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
−9.071761678372469580879485959208, −8.485649624601831563190444773773, −7.37375962182415421915695877316, −6.95964162904094422467518638377, −5.87971616260330106800847956364, −4.72823235796080818898614285768, −4.27298702203300212674995424033, −3.04431888294408595787500730825, −1.90355745214502556607500899144, −0.72949639089372181159234675690,
0.72949639089372181159234675690, 1.90355745214502556607500899144, 3.04431888294408595787500730825, 4.27298702203300212674995424033, 4.72823235796080818898614285768, 5.87971616260330106800847956364, 6.95964162904094422467518638377, 7.37375962182415421915695877316, 8.485649624601831563190444773773, 9.071761678372469580879485959208
