| L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s + 11-s + 3·15-s − 8·17-s + 6·19-s − 21-s + 23-s + 4·25-s − 27-s − 29-s − 33-s − 3·35-s + 6·37-s + 11·41-s − 11·43-s − 3·45-s − 12·47-s − 6·49-s + 8·51-s − 2·53-s − 3·55-s − 6·57-s − 7·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.774·15-s − 1.94·17-s + 1.37·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.185·29-s − 0.174·33-s − 0.507·35-s + 0.986·37-s + 1.71·41-s − 1.67·43-s − 0.447·45-s − 1.75·47-s − 6/7·49-s + 1.12·51-s − 0.274·53-s − 0.404·55-s − 0.794·57-s − 0.911·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−12.73334179301885, −12.01288189408761, −11.90251834003088, −11.28816990192677, −11.09656692330572, −10.86750407742445, −10.04246988485956, −9.419078359902378, −9.305068752317921, −8.500969806964064, −8.122416277547421, −7.759863937402264, −7.103618085471731, −6.944942011583758, −6.226299124613383, −5.883320584131426, −5.076145988638630, −4.601436790306398, −4.447753639586938, −3.822462649700727, −3.166850707538627, −2.808234813663747, −1.823627640266093, −1.404998245253311, −0.5562971271208447, 0,
0.5562971271208447, 1.404998245253311, 1.823627640266093, 2.808234813663747, 3.166850707538627, 3.822462649700727, 4.447753639586938, 4.601436790306398, 5.076145988638630, 5.883320584131426, 6.226299124613383, 6.944942011583758, 7.103618085471731, 7.759863937402264, 8.122416277547421, 8.500969806964064, 9.305068752317921, 9.419078359902378, 10.04246988485956, 10.86750407742445, 11.09656692330572, 11.28816990192677, 11.90251834003088, 12.01288189408761, 12.73334179301885
