| L(s) = 1 | − 2·5-s + 4·7-s + 6·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s − 8·35-s − 6·37-s − 10·41-s + 8·43-s + 9·49-s + 6·53-s − 4·59-s − 2·61-s − 12·65-s − 12·67-s − 8·71-s − 2·73-s − 4·79-s − 12·83-s − 12·85-s + 6·89-s + 24·91-s − 16·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.66·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.35·35-s − 0.986·37-s − 1.56·41-s + 1.21·43-s + 9/7·49-s + 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.450·79-s − 1.31·83-s − 1.30·85-s + 0.635·89-s + 2.51·91-s − 1.64·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.650337562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.650337562\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 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.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.13578383648410, −13.75077402444697, −13.35241423714668, −12.38778912460760, −11.97249968005020, −11.66005368699347, −11.33294611943417, −10.58786898290818, −10.32229507422756, −9.584449752292525, −8.786593303703411, −8.476465708637807, −8.007280257345233, −7.455006257720046, −7.227032321049431, −6.232751203762398, −5.615509925837473, −5.267630885104110, −4.534690148159228, −4.040836492337643, −3.294349742768924, −3.055900695743584, −1.743352181177247, −1.307331290978271, −0.7231006048604419,
0.7231006048604419, 1.307331290978271, 1.743352181177247, 3.055900695743584, 3.294349742768924, 4.040836492337643, 4.534690148159228, 5.267630885104110, 5.615509925837473, 6.232751203762398, 7.227032321049431, 7.455006257720046, 8.007280257345233, 8.476465708637807, 8.786593303703411, 9.584449752292525, 10.32229507422756, 10.58786898290818, 11.33294611943417, 11.66005368699347, 11.97249968005020, 12.38778912460760, 13.35241423714668, 13.75077402444697, 14.13578383648410
