| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 5·11-s + 16-s − 2·17-s + 19-s − 20-s + 5·22-s − 23-s − 4·25-s + 4·29-s + 9·31-s + 32-s − 2·34-s + 5·37-s + 38-s − 40-s + 9·41-s − 10·43-s + 5·44-s − 46-s − 6·47-s − 4·50-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.50·11-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.223·20-s + 1.06·22-s − 0.208·23-s − 4/5·25-s + 0.742·29-s + 1.61·31-s + 0.176·32-s − 0.342·34-s + 0.821·37-s + 0.162·38-s − 0.158·40-s + 1.40·41-s − 1.52·43-s + 0.753·44-s − 0.147·46-s − 0.875·47-s − 0.565·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.940488761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.940488761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.758419521307545992457221280358, −8.073840738987368495962393102681, −7.15629851020103523042213387688, −6.47784583599995832693494924500, −5.87450213364309522501266401702, −4.69069334189536458166730202416, −4.15202289907072344630083460584, −3.35511404251805033708538922757, −2.26551691900571842895890768283, −1.01936668222152272292011651270,
1.01936668222152272292011651270, 2.26551691900571842895890768283, 3.35511404251805033708538922757, 4.15202289907072344630083460584, 4.69069334189536458166730202416, 5.87450213364309522501266401702, 6.47784583599995832693494924500, 7.15629851020103523042213387688, 8.073840738987368495962393102681, 8.758419521307545992457221280358
