| L(s) = 1 | + 3·3-s − 2·4-s + 5-s + 6·9-s − 11-s − 6·12-s + 4·13-s + 3·15-s + 4·16-s − 2·17-s + 6·19-s − 2·20-s − 5·23-s − 4·25-s + 9·27-s + 10·29-s − 31-s − 3·33-s − 12·36-s − 5·37-s + 12·39-s + 2·41-s − 8·43-s + 2·44-s + 6·45-s − 8·47-s + 12·48-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 0.447·5-s + 2·9-s − 0.301·11-s − 1.73·12-s + 1.10·13-s + 0.774·15-s + 16-s − 0.485·17-s + 1.37·19-s − 0.447·20-s − 1.04·23-s − 4/5·25-s + 1.73·27-s + 1.85·29-s − 0.179·31-s − 0.522·33-s − 2·36-s − 0.821·37-s + 1.92·39-s + 0.312·41-s − 1.21·43-s + 0.301·44-s + 0.894·45-s − 1.16·47-s + 1.73·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278607672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278607672\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−10.29316588242584326072512226536, −9.771899647357342277831560950090, −8.947549675853731093244887653421, −8.333198695137771254103563674887, −7.66866498238269472650919646245, −6.27653205655726142636171995192, −4.95399628290113050953278162390, −3.85266175099960933742282200478, −3.04533793513601036750822233008, −1.58751343454368774869669820755,
1.58751343454368774869669820755, 3.04533793513601036750822233008, 3.85266175099960933742282200478, 4.95399628290113050953278162390, 6.27653205655726142636171995192, 7.66866498238269472650919646245, 8.333198695137771254103563674887, 8.947549675853731093244887653421, 9.771899647357342277831560950090, 10.29316588242584326072512226536
