| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 3·11-s + 15-s − 5·17-s + 2·19-s + 21-s + 3·23-s + 25-s + 27-s − 4·31-s + 3·33-s + 35-s + 37-s − 9·41-s + 2·43-s + 45-s + 8·47-s − 6·49-s − 5·51-s + 53-s + 3·55-s + 2·57-s − 4·59-s + 3·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.258·15-s − 1.21·17-s + 0.458·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s + 0.522·33-s + 0.169·35-s + 0.164·37-s − 1.40·41-s + 0.304·43-s + 0.149·45-s + 1.16·47-s − 6/7·49-s − 0.700·51-s + 0.137·53-s + 0.404·55-s + 0.264·57-s − 0.520·59-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.503816170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.503816170\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−13.39288793502515, −12.80904276951690, −12.44311702614956, −11.77248908038133, −11.37715421375739, −10.88248609518023, −10.45792954404211, −9.724842448368674, −9.447121866801789, −8.842480612157628, −8.676682856056424, −7.992095835925530, −7.391462984850639, −6.965373730201746, −6.446608957120209, −6.017809783239204, −5.180411677262677, −4.858305134910856, −4.251816036753998, −3.560151980815126, −3.258526719889593, −2.205105275705152, −2.105374420190614, −1.275896451636024, −0.6105128843795583,
0.6105128843795583, 1.275896451636024, 2.105374420190614, 2.205105275705152, 3.258526719889593, 3.560151980815126, 4.251816036753998, 4.858305134910856, 5.180411677262677, 6.017809783239204, 6.446608957120209, 6.965373730201746, 7.391462984850639, 7.992095835925530, 8.676682856056424, 8.842480612157628, 9.447121866801789, 9.724842448368674, 10.45792954404211, 10.88248609518023, 11.37715421375739, 11.77248908038133, 12.44311702614956, 12.80904276951690, 13.39288793502515
