| L(s) = 1 | + 3·3-s − 5-s + 7-s + 6·9-s + 11-s − 13-s − 3·15-s − 3·17-s + 8·19-s + 3·21-s + 4·23-s + 25-s + 9·27-s + 3·29-s − 6·31-s + 3·33-s − 35-s − 8·37-s − 3·39-s + 10·41-s − 12·43-s − 6·45-s + 3·47-s + 49-s − 9·51-s + 12·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 0.301·11-s − 0.277·13-s − 0.774·15-s − 0.727·17-s + 1.83·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s − 1.07·31-s + 0.522·33-s − 0.169·35-s − 1.31·37-s − 0.480·39-s + 1.56·41-s − 1.82·43-s − 0.894·45-s + 0.437·47-s + 1/7·49-s − 1.26·51-s + 1.64·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.972427655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.972427655\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| 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
−9.553827395506350423515858298763, −8.903289819479964213274609398601, −8.343262566908770294029950958935, −7.34391358243048062613743785964, −7.06383571809455944775407314183, −5.37537592180299819169889524025, −4.34052709019833991060180339015, −3.45731619098418144055547198795, −2.65914783079728487010329876439, −1.43027277813161202559840778497,
1.43027277813161202559840778497, 2.65914783079728487010329876439, 3.45731619098418144055547198795, 4.34052709019833991060180339015, 5.37537592180299819169889524025, 7.06383571809455944775407314183, 7.34391358243048062613743785964, 8.343262566908770294029950958935, 8.903289819479964213274609398601, 9.553827395506350423515858298763
