| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 3·11-s + 2·13-s + 3·15-s − 3·17-s − 19-s + 3·23-s + 4·25-s − 5·27-s + 6·29-s + 7·31-s + 3·33-s + 37-s + 2·39-s − 6·41-s + 4·43-s − 6·45-s + 9·47-s − 3·51-s − 3·53-s + 9·55-s − 57-s + 9·59-s − 61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.904·11-s + 0.554·13-s + 0.774·15-s − 0.727·17-s − 0.229·19-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 1.11·29-s + 1.25·31-s + 0.522·33-s + 0.164·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.894·45-s + 1.31·47-s − 0.420·51-s − 0.412·53-s + 1.21·55-s − 0.132·57-s + 1.17·59-s − 0.128·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083635063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083635063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.726570655503041576078296926786, −8.268150639737890634390598413755, −7.01623711877011981480625383115, −6.35716305946427278151931218063, −5.81955452643951491633087958766, −4.86830628267247308802282910099, −3.90109633321229907531052507392, −2.85801302383608422743649584746, −2.18409302870715836807119647030, −1.10116503864114736537536395657,
1.10116503864114736537536395657, 2.18409302870715836807119647030, 2.85801302383608422743649584746, 3.90109633321229907531052507392, 4.86830628267247308802282910099, 5.81955452643951491633087958766, 6.35716305946427278151931218063, 7.01623711877011981480625383115, 8.268150639737890634390598413755, 8.726570655503041576078296926786
