| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·11-s − 4·13-s − 15-s − 2·17-s − 19-s + 2·21-s − 4·23-s + 25-s + 27-s + 8·31-s + 2·33-s − 2·35-s − 8·37-s − 4·39-s − 8·41-s − 6·43-s − 45-s + 12·47-s − 3·49-s − 2·51-s + 6·53-s − 2·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.348·33-s − 0.338·35-s − 1.31·37-s − 0.640·39-s − 1.24·41-s − 0.914·43-s − 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−15.87015500083846, −15.35526118037101, −15.03876428771849, −14.42681638272214, −13.83208089485127, −13.62754658657594, −12.60930655671823, −12.11004605353570, −11.81102113790751, −11.10031050204416, −10.38145158011893, −9.943006485520868, −9.225963711665705, −8.626904568169735, −8.100773926233191, −7.688702607912627, −6.749816856314759, −6.615530936928657, −5.332760017807483, −4.927313745114399, −4.126574560834554, −3.697664945568671, −2.663393399353302, −2.098206588134155, −1.226774211350724, 0,
1.226774211350724, 2.098206588134155, 2.663393399353302, 3.697664945568671, 4.126574560834554, 4.927313745114399, 5.332760017807483, 6.615530936928657, 6.749816856314759, 7.688702607912627, 8.100773926233191, 8.626904568169735, 9.225963711665705, 9.943006485520868, 10.38145158011893, 11.10031050204416, 11.81102113790751, 12.11004605353570, 12.60930655671823, 13.62754658657594, 13.83208089485127, 14.42681638272214, 15.03876428771849, 15.35526118037101, 15.87015500083846
