| L(s) = 1 | + 2·7-s + 2·11-s − 2·13-s − 2·17-s − 8·19-s − 5·25-s − 6·29-s − 8·31-s − 10·37-s − 10·41-s + 4·43-s − 8·47-s − 3·49-s + 12·53-s − 12·59-s − 6·61-s − 12·67-s − 12·71-s − 6·73-s + 4·77-s + 14·79-s + 6·83-s + 14·89-s − 4·91-s + 16·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 25-s − 1.11·29-s − 1.43·31-s − 1.64·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 1.64·53-s − 1.56·59-s − 0.768·61-s − 1.46·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s + 1.57·79-s + 0.658·83-s + 1.48·89-s − 0.419·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64413531659595, −13.40166708755578, −12.91899784545379, −12.18846125360825, −11.98749011585369, −11.47060808460353, −10.83384072602019, −10.59114159309461, −10.09022400025472, −9.321671292763331, −8.910562743563110, −8.686087615900129, −7.847617281013872, −7.603509839431152, −6.953705939317960, −6.449496970113601, −5.972749364978615, −5.294656399329324, −4.845799128582146, −4.279032596050375, −3.772854314205045, −3.257969109037619, −2.268190248994850, −1.829538834017357, −1.514066542034926, 0, 0,
1.514066542034926, 1.829538834017357, 2.268190248994850, 3.257969109037619, 3.772854314205045, 4.279032596050375, 4.845799128582146, 5.294656399329324, 5.972749364978615, 6.449496970113601, 6.953705939317960, 7.603509839431152, 7.847617281013872, 8.686087615900129, 8.910562743563110, 9.321671292763331, 10.09022400025472, 10.59114159309461, 10.83384072602019, 11.47060808460353, 11.98749011585369, 12.18846125360825, 12.91899784545379, 13.40166708755578, 13.64413531659595
