| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s − 4·11-s − 12-s + 2·13-s − 15-s − 16-s − 2·17-s + 18-s + 20-s − 4·22-s + 8·23-s − 3·24-s + 25-s + 2·26-s + 27-s + 10·29-s − 30-s + 4·31-s + 5·32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.223·20-s − 0.852·22-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.85·29-s − 0.182·30-s + 0.718·31-s + 0.883·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221368062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221368062\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 163 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−8.704287932522414175699846258413, −8.368497396096667917446742191983, −7.47738930493919395423432976680, −6.56562829396827391060331887508, −5.66241818835706235560024597077, −4.71596782682335781027103685553, −4.30500807572919370962051326769, −3.10223838202310795165601485529, −2.69411041556594394393594930744, −0.838773042190839979450102072513,
0.838773042190839979450102072513, 2.69411041556594394393594930744, 3.10223838202310795165601485529, 4.30500807572919370962051326769, 4.71596782682335781027103685553, 5.66241818835706235560024597077, 6.56562829396827391060331887508, 7.47738930493919395423432976680, 8.368497396096667917446742191983, 8.704287932522414175699846258413
