| L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s − 4·13-s − 14-s + 16-s + 6·17-s − 4·19-s + 3·20-s + 6·23-s + 4·25-s + 4·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 3·35-s − 7·37-s + 4·38-s − 3·40-s − 3·41-s − 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.670·20-s + 1.25·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.507·35-s − 1.15·37-s + 0.648·38-s − 0.474·40-s − 0.468·41-s − 0.609·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.164616788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.164616788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
| 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
−14.66452787541156, −14.13274118240064, −13.75221349160160, −12.91730876462715, −12.66451762234145, −11.93888108581046, −11.58581026258250, −10.68060761911095, −10.30629734681093, −10.02381378364961, −9.457547602880900, −8.763195346852300, −8.562725111982653, −7.658995258914282, −7.194906540682268, −6.704996449180560, −5.974179660515047, −5.468850425364392, −4.998684556592367, −4.293877730820793, −3.125199752129334, −2.827290636232570, −1.844738919951989, −1.580327789429453, −0.5788379060379778,
0.5788379060379778, 1.580327789429453, 1.844738919951989, 2.827290636232570, 3.125199752129334, 4.293877730820793, 4.998684556592367, 5.468850425364392, 5.974179660515047, 6.704996449180560, 7.194906540682268, 7.658995258914282, 8.562725111982653, 8.763195346852300, 9.457547602880900, 10.02381378364961, 10.30629734681093, 10.68060761911095, 11.58581026258250, 11.93888108581046, 12.66451762234145, 12.91730876462715, 13.75221349160160, 14.13274118240064, 14.66452787541156
