| L(s) = 1 | − 5-s − 2·7-s + 4·11-s + 3·13-s + 6·17-s + 2·19-s − 4·25-s + 5·29-s − 2·31-s + 2·35-s − 10·37-s − 3·41-s − 6·43-s − 3·49-s + 3·53-s − 4·55-s − 6·59-s − 3·61-s − 3·65-s − 4·67-s − 6·71-s − 73-s − 8·77-s − 8·79-s − 6·83-s − 6·85-s + 3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.832·13-s + 1.45·17-s + 0.458·19-s − 4/5·25-s + 0.928·29-s − 0.359·31-s + 0.338·35-s − 1.64·37-s − 0.468·41-s − 0.914·43-s − 3/7·49-s + 0.412·53-s − 0.539·55-s − 0.781·59-s − 0.384·61-s − 0.372·65-s − 0.488·67-s − 0.712·71-s − 0.117·73-s − 0.911·77-s − 0.900·79-s − 0.658·83-s − 0.650·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.81340897703529, −12.33372558986203, −11.96643207488595, −11.65566155015004, −11.22433384818801, −10.48502486260037, −10.12030725203259, −9.787700965358832, −9.189152058152148, −8.759080344419012, −8.359979992711203, −7.788475759914035, −7.268203836991755, −6.855380265899793, −6.312145761925753, −5.925747105679359, −5.425317387906230, −4.787497630375793, −4.161676714657773, −3.655206719234142, −3.261487751885488, −2.989986902760363, −1.810411810638836, −1.474581154070883, −0.7867787905133723, 0,
0.7867787905133723, 1.474581154070883, 1.810411810638836, 2.989986902760363, 3.261487751885488, 3.655206719234142, 4.161676714657773, 4.787497630375793, 5.425317387906230, 5.925747105679359, 6.312145761925753, 6.855380265899793, 7.268203836991755, 7.788475759914035, 8.359979992711203, 8.759080344419012, 9.189152058152148, 9.787700965358832, 10.12030725203259, 10.48502486260037, 11.22433384818801, 11.65566155015004, 11.96643207488595, 12.33372558986203, 12.81340897703529
