| L(s) = 1 | − 2·5-s + 4·7-s − 3·9-s + 11-s + 13-s − 6·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 8·31-s − 8·35-s − 2·37-s + 6·41-s + 4·43-s + 6·45-s − 8·47-s + 9·49-s + 2·53-s − 2·55-s + 12·59-s + 14·61-s − 12·63-s − 2·65-s + 12·67-s + 6·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 9-s + 0.301·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 1.35·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.79·61-s − 1.51·63-s − 0.248·65-s + 1.46·67-s + 0.702·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284226189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284226189\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.015362656270658305898360804147, −7.08990868052285484694574468914, −6.38732402858569428318342292360, −5.61340628516362643719882493293, −4.88493698071684294680056988451, −4.10581968299170667261122740052, −3.77548951204172391347859981026, −2.38716332269974273854531335130, −1.92302789878394110015929556110, −0.52687804293475321840598315593,
0.52687804293475321840598315593, 1.92302789878394110015929556110, 2.38716332269974273854531335130, 3.77548951204172391347859981026, 4.10581968299170667261122740052, 4.88493698071684294680056988451, 5.61340628516362643719882493293, 6.38732402858569428318342292360, 7.08990868052285484694574468914, 8.015362656270658305898360804147
