| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 13-s − 15-s − 6·17-s + 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s − 29-s + 8·31-s + 4·33-s + 4·35-s + 6·37-s − 39-s − 2·41-s + 4·43-s − 45-s + 9·49-s − 6·51-s + 10·53-s − 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.840·51-s + 1.37·53-s − 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.841962918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.841962918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.66978218325471, −11.95922751048360, −11.61011525221364, −11.48970142883362, −10.49264379449074, −10.17084646857274, −9.727885643091783, −9.361552809330974, −8.932267132903384, −8.514101052395134, −7.925081405155723, −7.491378051958935, −6.820807230013759, −6.606877297888649, −6.215088996130506, −5.620613263530568, −4.891153304142779, −4.203066774872060, −3.944441625314745, −3.556383533689028, −2.886228814974733, −2.378984181699069, −1.930756897377769, −0.8605783229031451, −0.5317289855231084,
0.5317289855231084, 0.8605783229031451, 1.930756897377769, 2.378984181699069, 2.886228814974733, 3.556383533689028, 3.944441625314745, 4.203066774872060, 4.891153304142779, 5.620613263530568, 6.215088996130506, 6.606877297888649, 6.820807230013759, 7.491378051958935, 7.925081405155723, 8.514101052395134, 8.932267132903384, 9.361552809330974, 9.727885643091783, 10.17084646857274, 10.49264379449074, 11.48970142883362, 11.61011525221364, 11.95922751048360, 12.66978218325471
