| L(s) = 1 | + 2.98·5-s + 11.7·7-s − 55.2·11-s − 5.58·13-s + 35.9·17-s + 107.·19-s + 162.·23-s − 116.·25-s + 15.2·29-s + 108.·31-s + 34.9·35-s + 415.·37-s − 57.5·41-s − 178.·43-s − 463.·47-s − 205.·49-s − 234.·53-s − 164.·55-s + 199.·59-s + 849.·61-s − 16.6·65-s + 1.04e3·67-s + 533.·71-s + 477.·73-s − 646.·77-s + 1.05e3·79-s + 1.05e3·83-s + ⋯ |
| L(s) = 1 | + 0.266·5-s + 0.632·7-s − 1.51·11-s − 0.119·13-s + 0.513·17-s + 1.29·19-s + 1.47·23-s − 0.928·25-s + 0.0974·29-s + 0.628·31-s + 0.168·35-s + 1.84·37-s − 0.219·41-s − 0.631·43-s − 1.43·47-s − 0.600·49-s − 0.608·53-s − 0.403·55-s + 0.440·59-s + 1.78·61-s − 0.0317·65-s + 1.90·67-s + 0.892·71-s + 0.765·73-s − 0.957·77-s + 1.50·79-s + 1.40·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.160169536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.160169536\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.98T + 125T^{2} \) |
| 7 | \( 1 - 11.7T + 343T^{2} \) |
| 11 | \( 1 + 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.58T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 415.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 57.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 199.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 849.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 533.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 608.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 263.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03708005190716490816350564656, −9.502724403685551016504425471806, −8.115166658317102954283380564239, −7.79186091476727055704247220142, −6.60797987407613783602699337965, −5.33243951046120362126819571890, −4.94453313798705587740909735806, −3.35935394010443121552335004229, −2.31398522638731363665142231305, −0.879339557205483972281069867134,
0.879339557205483972281069867134, 2.31398522638731363665142231305, 3.35935394010443121552335004229, 4.94453313798705587740909735806, 5.33243951046120362126819571890, 6.60797987407613783602699337965, 7.79186091476727055704247220142, 8.115166658317102954283380564239, 9.502724403685551016504425471806, 10.03708005190716490816350564656
