| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 11-s + 2·13-s − 17-s + 19-s − 6·21-s + 4·23-s − 9·27-s + 2·29-s − 4·31-s + 3·33-s − 6·37-s − 6·39-s − 2·41-s − 5·43-s + 2·47-s − 3·49-s + 3·51-s − 2·53-s − 3·57-s + 7·61-s + 12·63-s − 14·67-s − 12·69-s + 15·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 0.301·11-s + 0.554·13-s − 0.242·17-s + 0.229·19-s − 1.30·21-s + 0.834·23-s − 1.73·27-s + 0.371·29-s − 0.718·31-s + 0.522·33-s − 0.986·37-s − 0.960·39-s − 0.312·41-s − 0.762·43-s + 0.291·47-s − 3/7·49-s + 0.420·51-s − 0.274·53-s − 0.397·57-s + 0.896·61-s + 1.51·63-s − 1.71·67-s − 1.44·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408674940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408674940\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 229 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−12.19180515144657, −12.13069988872604, −11.52417435534082, −11.10340278210298, −10.87798630105594, −10.44678998546235, −10.00859977716266, −9.393184845936776, −8.916012201236216, −8.334174849206088, −7.875692333275138, −7.326984320797079, −6.761554104011176, −6.572983054998609, −5.913751637502064, −5.414828457461210, −5.090570480745651, −4.738592610055829, −4.155043418374765, −3.547450028844477, −2.971947578627522, −1.987427720516492, −1.641779811353001, −0.9257706040643816, −0.4233302162916470,
0.4233302162916470, 0.9257706040643816, 1.641779811353001, 1.987427720516492, 2.971947578627522, 3.547450028844477, 4.155043418374765, 4.738592610055829, 5.090570480745651, 5.414828457461210, 5.913751637502064, 6.572983054998609, 6.761554104011176, 7.326984320797079, 7.875692333275138, 8.334174849206088, 8.916012201236216, 9.393184845936776, 10.00859977716266, 10.44678998546235, 10.87798630105594, 11.10340278210298, 11.52417435534082, 12.13069988872604, 12.19180515144657
