| L(s) = 1 | − 2-s + 4-s + 4·5-s − 2·7-s − 8-s − 4·10-s − 11-s + 4·13-s + 2·14-s + 16-s + 2·17-s + 4·20-s + 22-s + 6·23-s + 11·25-s − 4·26-s − 2·28-s − 10·29-s − 8·31-s − 32-s − 2·34-s − 8·35-s − 2·37-s − 4·40-s − 2·41-s + 4·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.755·7-s − 0.353·8-s − 1.26·10-s − 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.894·20-s + 0.213·22-s + 1.25·23-s + 11/5·25-s − 0.784·26-s − 0.377·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.35·35-s − 0.328·37-s − 0.632·40-s − 0.312·41-s + 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085741366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085741366\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−12.78754919426025536298321895292, −11.10629631807433205694642872180, −10.38013514973553904358785803492, −9.396010671988393299929043424642, −8.935424921700037406617751999461, −7.31626404793277731789533693062, −6.20004880727144301320499647411, −5.48462700120923997068508135767, −3.18510621994423060681354856306, −1.66815280525216283373919758824,
1.66815280525216283373919758824, 3.18510621994423060681354856306, 5.48462700120923997068508135767, 6.20004880727144301320499647411, 7.31626404793277731789533693062, 8.935424921700037406617751999461, 9.396010671988393299929043424642, 10.38013514973553904358785803492, 11.10629631807433205694642872180, 12.78754919426025536298321895292
