| L(s) = 1 | − 3-s − 2·7-s + 9-s + 3·11-s + 17-s + 19-s + 2·21-s + 3·23-s − 27-s − 7·29-s − 7·31-s − 3·33-s + 6·41-s + 4·43-s + 4·47-s − 3·49-s − 51-s + 5·53-s − 57-s + 12·59-s + 11·61-s − 2·63-s + 11·67-s − 3·69-s − 12·71-s − 5·73-s − 6·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.242·17-s + 0.229·19-s + 0.436·21-s + 0.625·23-s − 0.192·27-s − 1.29·29-s − 1.25·31-s − 0.522·33-s + 0.937·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.140·51-s + 0.686·53-s − 0.132·57-s + 1.56·59-s + 1.40·61-s − 0.251·63-s + 1.34·67-s − 0.361·69-s − 1.42·71-s − 0.585·73-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146046147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146046147\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.46392328327472, −12.03034490927686, −11.41265032939398, −11.28173094563271, −10.68483897478185, −10.27025146151716, −9.558701966462923, −9.441903374917420, −9.029621892962092, −8.373107660672826, −7.865305736480770, −7.179417872365596, −6.917657274920881, −6.625615016357672, −5.747487382067420, −5.619467111578720, −5.216013475742180, −4.287602114572739, −3.930013910033718, −3.642411262080938, −2.811308723770906, −2.389475969616947, −1.503997564051469, −1.117165026293593, −0.3091673173811771,
0.3091673173811771, 1.117165026293593, 1.503997564051469, 2.389475969616947, 2.811308723770906, 3.642411262080938, 3.930013910033718, 4.287602114572739, 5.216013475742180, 5.619467111578720, 5.747487382067420, 6.625615016357672, 6.917657274920881, 7.179417872365596, 7.865305736480770, 8.373107660672826, 9.029621892962092, 9.441903374917420, 9.558701966462923, 10.27025146151716, 10.68483897478185, 11.28173094563271, 11.41265032939398, 12.03034490927686, 12.46392328327472
