| L(s) = 1 | − 3-s + 5-s + 2·7-s − 2·9-s + 4·11-s − 13-s − 15-s + 17-s − 19-s − 2·21-s + 6·23-s + 25-s + 5·27-s − 5·29-s + 3·31-s − 4·33-s + 2·35-s + 4·37-s + 39-s + 6·41-s + 10·43-s − 2·45-s + 5·47-s − 3·49-s − 51-s − 3·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 0.538·31-s − 0.696·33-s + 0.338·35-s + 0.657·37-s + 0.160·39-s + 0.937·41-s + 1.52·43-s − 0.298·45-s + 0.729·47-s − 3/7·49-s − 0.140·51-s − 0.412·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.466377890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466377890\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76792248171826519581251782244, −9.502292028355170162692614931059, −8.922867142689585025777010776290, −7.87213974837717368652492799178, −6.81754514899596852986711667831, −5.94662038287314899491975310750, −5.14884523236919836804552962057, −4.11049804585327936028225219196, −2.63271151351926318644565002461, −1.15493306207939279826165810518,
1.15493306207939279826165810518, 2.63271151351926318644565002461, 4.11049804585327936028225219196, 5.14884523236919836804552962057, 5.94662038287314899491975310750, 6.81754514899596852986711667831, 7.87213974837717368652492799178, 8.922867142689585025777010776290, 9.502292028355170162692614931059, 10.76792248171826519581251782244
