| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 4·13-s + 16-s + 6·17-s + 4·19-s + 3·20-s − 6·23-s + 4·25-s − 4·26-s + 3·29-s − 8·31-s − 32-s − 6·34-s + 8·37-s − 4·38-s − 3·40-s − 6·41-s + 8·43-s + 6·46-s + 6·47-s − 4·50-s + 4·52-s − 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s − 0.784·26-s + 0.557·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.31·37-s − 0.648·38-s − 0.474·40-s − 0.937·41-s + 1.21·43-s + 0.884·46-s + 0.875·47-s − 0.565·50-s + 0.554·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.934749980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934749980\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−9.080307510754945695338223626417, −8.089162467985480210563653789876, −7.53555940761379996453814684424, −6.44746236575269500243043966102, −5.85181888271394769605901931939, −5.34295331715232478556650999545, −3.89746653429416377815915424510, −2.94211773623874109816859654251, −1.86269570359270794591117296706, −1.05404344222439955836076699282,
1.05404344222439955836076699282, 1.86269570359270794591117296706, 2.94211773623874109816859654251, 3.89746653429416377815915424510, 5.34295331715232478556650999545, 5.85181888271394769605901931939, 6.44746236575269500243043966102, 7.53555940761379996453814684424, 8.089162467985480210563653789876, 9.080307510754945695338223626417
