| L(s) = 1 | + 5-s + 7-s + 4·11-s − 13-s − 6·17-s − 8·19-s − 4·23-s + 25-s + 2·29-s + 4·31-s + 35-s + 10·37-s + 2·41-s − 8·43-s + 8·47-s + 49-s − 6·53-s + 4·55-s + 8·59-s − 10·61-s − 65-s + 12·67-s − 8·71-s + 6·73-s + 4·77-s − 8·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.169·35-s + 1.64·37-s + 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 1.04·59-s − 1.28·61-s − 0.124·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.45076427812126, −14.06649451467602, −13.43712648916285, −12.95856496311062, −12.57325466920423, −11.75477649241920, −11.59310887415537, −10.88228170702583, −10.47025735265512, −9.892593106688619, −9.253537909610125, −8.933329486873597, −8.276439829201188, −7.962148200150533, −6.968726056455402, −6.604869209938298, −6.214885699447549, −5.642678746977033, −4.707992378365140, −4.309127431356571, −4.006219004702724, −2.940877493168380, −2.244801650358576, −1.861077719332282, −0.9820166049135922, 0,
0.9820166049135922, 1.861077719332282, 2.244801650358576, 2.940877493168380, 4.006219004702724, 4.309127431356571, 4.707992378365140, 5.642678746977033, 6.214885699447549, 6.604869209938298, 6.968726056455402, 7.962148200150533, 8.276439829201188, 8.933329486873597, 9.253537909610125, 9.892593106688619, 10.47025735265512, 10.88228170702583, 11.59310887415537, 11.75477649241920, 12.57325466920423, 12.95856496311062, 13.43712648916285, 14.06649451467602, 14.45076427812126
