| L(s) = 1 | + 3·5-s + 7-s − 3·11-s + 4·13-s − 6·17-s + 2·19-s + 3·23-s + 4·25-s + 2·31-s + 3·35-s + 9·37-s − 9·41-s − 2·43-s + 7·47-s − 6·49-s + 53-s − 9·55-s − 10·59-s − 4·61-s + 12·65-s + 17·73-s − 3·77-s + 8·79-s − 3·83-s − 18·85-s − 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.904·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s + 0.625·23-s + 4/5·25-s + 0.359·31-s + 0.507·35-s + 1.47·37-s − 1.40·41-s − 0.304·43-s + 1.02·47-s − 6/7·49-s + 0.137·53-s − 1.21·55-s − 1.30·59-s − 0.512·61-s + 1.48·65-s + 1.98·73-s − 0.341·77-s + 0.900·79-s − 0.329·83-s − 1.95·85-s − 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.518381049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.518381049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.02505021331522, −12.68730393824241, −12.01399089312960, −11.34125082883014, −11.00718016584255, −10.72862299170831, −10.08248139576918, −9.727320374200156, −9.124207643186962, −8.854325180886089, −8.208538811710351, −7.875643539731267, −7.158419862659185, −6.556008243531805, −6.275568751562327, −5.743521873058355, −5.188916845809318, −4.822496720596260, −4.238324705324366, −3.513335936067074, −2.835095768495834, −2.420127570369017, −1.785724618522525, −1.313305130362830, −0.5084439608641940,
0.5084439608641940, 1.313305130362830, 1.785724618522525, 2.420127570369017, 2.835095768495834, 3.513335936067074, 4.238324705324366, 4.822496720596260, 5.188916845809318, 5.743521873058355, 6.275568751562327, 6.556008243531805, 7.158419862659185, 7.875643539731267, 8.208538811710351, 8.854325180886089, 9.124207643186962, 9.727320374200156, 10.08248139576918, 10.72862299170831, 11.00718016584255, 11.34125082883014, 12.01399089312960, 12.68730393824241, 13.02505021331522
