| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 2·11-s − 2·13-s + 2·15-s − 4·17-s − 2·21-s − 23-s − 25-s − 27-s − 10·29-s + 2·33-s − 4·35-s − 4·37-s + 2·39-s − 6·41-s − 4·43-s − 2·45-s + 8·47-s − 3·49-s + 4·51-s − 6·53-s + 4·55-s + 4·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.516·15-s − 0.970·17-s − 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.348·33-s − 0.676·35-s − 0.657·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−10.60170946307074366403508413309, −9.526038370460309422774263796566, −8.391316645228763022524368582949, −7.64933333759264638051491407709, −6.85071419727856154486285530980, −5.52763608293326608873748556277, −4.72068079232042573957766086280, −3.71619350236108323871179096178, −2.03589601036470946350610644421, 0,
2.03589601036470946350610644421, 3.71619350236108323871179096178, 4.72068079232042573957766086280, 5.52763608293326608873748556277, 6.85071419727856154486285530980, 7.64933333759264638051491407709, 8.391316645228763022524368582949, 9.526038370460309422774263796566, 10.60170946307074366403508413309
