| L(s) = 1 | + 2·3-s + 2·5-s + 9-s − 11-s − 4·13-s + 4·15-s − 4·17-s + 4·23-s − 25-s − 4·27-s − 6·29-s + 10·31-s − 2·33-s − 6·37-s − 8·39-s − 4·41-s − 12·43-s + 2·45-s − 10·47-s − 8·51-s − 6·53-s − 2·55-s + 2·59-s − 8·65-s − 8·67-s + 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.03·15-s − 0.970·17-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s + 1.79·31-s − 0.348·33-s − 0.986·37-s − 1.28·39-s − 0.624·41-s − 1.82·43-s + 0.298·45-s − 1.45·47-s − 1.12·51-s − 0.824·53-s − 0.269·55-s + 0.260·59-s − 0.992·65-s − 0.977·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 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
−7.51633165496452066565735742912, −6.78596846568392118954662100559, −6.18749203490606167640127654989, −5.13699302665429628170294703719, −4.79415566372924421890047210418, −3.62980781323120854331945127407, −2.94439912019830745165849006977, −2.24215446749499935575895860873, −1.67915649184277622840650438512, 0,
1.67915649184277622840650438512, 2.24215446749499935575895860873, 2.94439912019830745165849006977, 3.62980781323120854331945127407, 4.79415566372924421890047210418, 5.13699302665429628170294703719, 6.18749203490606167640127654989, 6.78596846568392118954662100559, 7.51633165496452066565735742912
