| L(s) = 1 | + 2·2-s + 4·4-s − 18.3·5-s − 7·7-s + 8·8-s − 36.7·10-s + 53.1·11-s + 7.22·13-s − 14·14-s + 16·16-s + 88·17-s − 134.·19-s − 73.5·20-s + 106.·22-s + 155.·23-s + 213.·25-s + 14.4·26-s − 28·28-s + 119.·29-s + 321.·31-s + 32·32-s + 176·34-s + 128.·35-s + 87.7·37-s − 268.·38-s − 147.·40-s − 226.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.64·5-s − 0.377·7-s + 0.353·8-s − 1.16·10-s + 1.45·11-s + 0.154·13-s − 0.267·14-s + 0.250·16-s + 1.25·17-s − 1.61·19-s − 0.822·20-s + 1.03·22-s + 1.40·23-s + 1.70·25-s + 0.108·26-s − 0.188·28-s + 0.765·29-s + 1.86·31-s + 0.176·32-s + 0.887·34-s + 0.621·35-s + 0.390·37-s − 1.14·38-s − 0.581·40-s − 0.863·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.301702797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301702797\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 18.3T + 125T^{2} \) |
| 11 | \( 1 - 53.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.22T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 321.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 87.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 62.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 382.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 220.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 813.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 262.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 455.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 414.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 17.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.49e3T + 9.12e5T^{2} \) |
| 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
−11.23003215161683760846463547506, −10.27824740478306645504110590392, −8.896264806800846180478561626020, −8.071607124129433189462564016197, −6.98787538009114119201407795352, −6.28858966332731088170025725793, −4.69221682946840315566274369504, −3.93229887483665029232425832570, −3.04420783862828029329497288967, −0.941632509186399643468946861725,
0.941632509186399643468946861725, 3.04420783862828029329497288967, 3.93229887483665029232425832570, 4.69221682946840315566274369504, 6.28858966332731088170025725793, 6.98787538009114119201407795352, 8.071607124129433189462564016197, 8.896264806800846180478561626020, 10.27824740478306645504110590392, 11.23003215161683760846463547506
