| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 5·13-s + 15-s − 7·17-s − 4·21-s − 4·25-s − 27-s − 7·29-s + 8·31-s − 4·35-s − 5·37-s − 5·39-s − 11·41-s − 8·43-s − 45-s − 4·47-s + 9·49-s + 7·51-s − 5·53-s + 2·61-s + 4·63-s − 5·65-s − 12·67-s + 16·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.38·13-s + 0.258·15-s − 1.69·17-s − 0.872·21-s − 4/5·25-s − 0.192·27-s − 1.29·29-s + 1.43·31-s − 0.676·35-s − 0.821·37-s − 0.800·39-s − 1.71·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.980·51-s − 0.686·53-s + 0.256·61-s + 0.503·63-s − 0.620·65-s − 1.46·67-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−7.939553874523233269343591707743, −6.92695649928071153214610786147, −6.41457237275786342055390731713, −5.50552820623364408561565638234, −4.82232746688580817899899121282, −4.21651525426244290111614742874, −3.45360462225753256585669593121, −2.02398060393285553597138892552, −1.41300408080002748308440869930, 0,
1.41300408080002748308440869930, 2.02398060393285553597138892552, 3.45360462225753256585669593121, 4.21651525426244290111614742874, 4.82232746688580817899899121282, 5.50552820623364408561565638234, 6.41457237275786342055390731713, 6.92695649928071153214610786147, 7.939553874523233269343591707743
