| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 13-s − 15-s + 5·19-s + 21-s − 23-s − 4·25-s + 27-s − 2·29-s − 6·31-s + 33-s − 35-s + 8·37-s + 39-s − 5·41-s − 43-s − 45-s + 2·47-s + 49-s + 6·53-s − 55-s + 5·57-s − 2·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.174·33-s − 0.169·35-s + 1.31·37-s + 0.160·39-s − 0.780·41-s − 0.152·43-s − 0.149·45-s + 0.291·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s + 0.662·57-s − 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.610830241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.610830241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| 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
−12.39589474948214, −11.94189676493878, −11.50894840559738, −11.29640976208968, −10.68322866722960, −10.00048401247533, −9.858653392139521, −9.232876221749582, −8.715369124864284, −8.503588389654663, −7.777285617342732, −7.482836965847782, −7.189810903672011, −6.539269640250375, −5.773489963021434, −5.646261815742572, −4.920015633633503, −4.238913183456762, −4.042889207743044, −3.391467554820058, −2.964702095969475, −2.324243759651774, −1.634569546442062, −1.257545194829259, −0.3989030031322454,
0.3989030031322454, 1.257545194829259, 1.634569546442062, 2.324243759651774, 2.964702095969475, 3.391467554820058, 4.042889207743044, 4.238913183456762, 4.920015633633503, 5.646261815742572, 5.773489963021434, 6.539269640250375, 7.189810903672011, 7.482836965847782, 7.777285617342732, 8.503588389654663, 8.715369124864284, 9.232876221749582, 9.858653392139521, 10.00048401247533, 10.68322866722960, 11.29640976208968, 11.50894840559738, 11.94189676493878, 12.39589474948214
