| L(s) = 1 | + 4·7-s − 2·11-s − 13-s − 6·17-s − 4·19-s + 4·23-s − 6·29-s + 8·31-s − 10·37-s + 4·41-s − 4·43-s − 6·47-s + 9·49-s − 6·53-s − 6·59-s + 6·61-s − 10·71-s + 2·73-s − 8·77-s + 10·83-s − 8·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.624·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.781·59-s + 0.768·61-s − 1.18·71-s + 0.234·73-s − 0.911·77-s + 1.09·83-s − 0.847·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259054946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259054946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.20822062746766, −12.73999485583376, −11.99655554433225, −11.80886765168895, −11.01826244929598, −10.88840376625418, −10.56913554390578, −9.781273570070177, −9.294433888938610, −8.695659174814131, −8.355324294596180, −7.968660028508102, −7.411735102801696, −6.792070560673466, −6.504261464967337, −5.675929644000214, −5.174208399714609, −4.736105884225135, −4.397397238243156, −3.761600008715965, −2.915580391065236, −2.408774464621109, −1.802979806036342, −1.378655123951677, −0.3045655119492183,
0.3045655119492183, 1.378655123951677, 1.802979806036342, 2.408774464621109, 2.915580391065236, 3.761600008715965, 4.397397238243156, 4.736105884225135, 5.174208399714609, 5.675929644000214, 6.504261464967337, 6.792070560673466, 7.411735102801696, 7.968660028508102, 8.355324294596180, 8.695659174814131, 9.294433888938610, 9.781273570070177, 10.56913554390578, 10.88840376625418, 11.01826244929598, 11.80886765168895, 11.99655554433225, 12.73999485583376, 13.20822062746766
