| L(s) = 1 | + 3·3-s − 4·5-s + 7-s + 6·9-s − 3·11-s + 2·13-s − 12·15-s + 8·17-s + 2·19-s + 3·21-s + 6·23-s + 11·25-s + 9·27-s + 8·31-s − 9·33-s − 4·35-s − 37-s + 6·39-s − 5·41-s + 2·43-s − 24·45-s − 11·47-s − 6·49-s + 24·51-s + 9·53-s + 12·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s − 0.904·11-s + 0.554·13-s − 3.09·15-s + 1.94·17-s + 0.458·19-s + 0.654·21-s + 1.25·23-s + 11/5·25-s + 1.73·27-s + 1.43·31-s − 1.56·33-s − 0.676·35-s − 0.164·37-s + 0.960·39-s − 0.780·41-s + 0.304·43-s − 3.57·45-s − 1.60·47-s − 6/7·49-s + 3.36·51-s + 1.23·53-s + 1.61·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.446585592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.446585592\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.648546434276898789747195507571, −8.490529166827239070163924840512, −8.217725009521119599085028706161, −7.63572304597459780773128038137, −6.98150467834209165101675556515, −5.20817284495310242430018493679, −4.27611685284512326935756014151, −3.29292662358312635823710209810, −2.97824748954438677044219419910, −1.19206411521547945511921007919,
1.19206411521547945511921007919, 2.97824748954438677044219419910, 3.29292662358312635823710209810, 4.27611685284512326935756014151, 5.20817284495310242430018493679, 6.98150467834209165101675556515, 7.63572304597459780773128038137, 8.217725009521119599085028706161, 8.490529166827239070163924840512, 9.648546434276898789747195507571
