| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 3·11-s + 2·13-s − 2·14-s + 16-s − 19-s + 3·22-s + 6·23-s − 5·25-s + 2·26-s − 2·28-s − 6·29-s + 4·31-s + 32-s + 4·37-s − 38-s − 9·41-s − 43-s + 3·44-s + 6·46-s − 6·47-s − 3·49-s − 5·50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.229·19-s + 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.657·37-s − 0.162·38-s − 1.40·41-s − 0.152·43-s + 0.452·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46818 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46818 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.675802565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.675802565\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−14.68773972922093, −13.95379322764582, −13.52639999429402, −13.08250689714908, −12.74874156213547, −11.86631541916583, −11.73239462271440, −11.08728943196990, −10.55130044558126, −9.842523448494835, −9.477094348105541, −8.826337785138418, −8.297606863474766, −7.544689560883724, −6.979659057614529, −6.408382374328885, −6.127481795462059, −5.358844083005151, −4.795753044838404, −4.018194957087718, −3.572936105975915, −3.085134889977817, −2.200500949598635, −1.509800830183413, −0.6090208166617369,
0.6090208166617369, 1.509800830183413, 2.200500949598635, 3.085134889977817, 3.572936105975915, 4.018194957087718, 4.795753044838404, 5.358844083005151, 6.127481795462059, 6.408382374328885, 6.979659057614529, 7.544689560883724, 8.297606863474766, 8.826337785138418, 9.477094348105541, 9.842523448494835, 10.55130044558126, 11.08728943196990, 11.73239462271440, 11.86631541916583, 12.74874156213547, 13.08250689714908, 13.52639999429402, 13.95379322764582, 14.68773972922093
