| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s + 17-s + 4·19-s − 4·23-s − 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·37-s + 2·39-s + 2·41-s + 8·43-s − 2·45-s + 51-s − 6·53-s + 8·55-s + 4·57-s − 4·59-s + 2·61-s − 4·65-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s + 0.140·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−13.63315238411716, −13.12258434903467, −12.42919341484588, −12.21760039472868, −11.64297536181205, −11.14824813055814, −10.65594932574796, −10.17535319285583, −9.681823501114233, −9.230952778668128, −8.535761741226618, −8.032644072618437, −7.873913084440818, −7.434768678783140, −6.774460987977125, −6.125999315945653, −5.641530870556149, −5.024735900218196, −4.365232190465762, −4.044306149944106, −3.321330003161221, −2.865648469219182, −2.394884914509595, −1.498393476446008, −0.8201295880497604, 0,
0.8201295880497604, 1.498393476446008, 2.394884914509595, 2.865648469219182, 3.321330003161221, 4.044306149944106, 4.365232190465762, 5.024735900218196, 5.641530870556149, 6.125999315945653, 6.774460987977125, 7.434768678783140, 7.873913084440818, 8.032644072618437, 8.535761741226618, 9.230952778668128, 9.681823501114233, 10.17535319285583, 10.65594932574796, 11.14824813055814, 11.64297536181205, 12.21760039472868, 12.42919341484588, 13.12258434903467, 13.63315238411716
