| L(s) = 1 | − 7-s − 3·11-s + 2·13-s + 6·17-s − 5·19-s + 9·23-s − 6·29-s − 31-s + 11·37-s + 3·41-s − 4·43-s + 12·47-s + 49-s − 8·61-s − 10·67-s + 3·71-s − 8·73-s + 3·77-s − 4·79-s + 6·83-s + 3·89-s − 2·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.904·11-s + 0.554·13-s + 1.45·17-s − 1.14·19-s + 1.87·23-s − 1.11·29-s − 0.179·31-s + 1.80·37-s + 0.468·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 1.02·61-s − 1.22·67-s + 0.356·71-s − 0.936·73-s + 0.341·77-s − 0.450·79-s + 0.658·83-s + 0.317·89-s − 0.209·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.633354481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.633354481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−12.69399907571201, −12.41505200404068, −11.74195715797750, −11.14584433658919, −10.93731876444265, −10.34657634448220, −10.09056053137855, −9.325671858902732, −9.088590784708814, −8.582983388727750, −7.959554216981782, −7.535514168410076, −7.241748217039765, −6.554099009279348, −5.929945169682986, −5.730316606299912, −5.155771880979353, −4.482510550071395, −4.130979997845448, −3.215877047618551, −3.142284238357699, −2.415189482500147, −1.772605279112797, −0.9952086200156859, −0.4988406075317932,
0.4988406075317932, 0.9952086200156859, 1.772605279112797, 2.415189482500147, 3.142284238357699, 3.215877047618551, 4.130979997845448, 4.482510550071395, 5.155771880979353, 5.730316606299912, 5.929945169682986, 6.554099009279348, 7.241748217039765, 7.535514168410076, 7.959554216981782, 8.582983388727750, 9.088590784708814, 9.325671858902732, 10.09056053137855, 10.34657634448220, 10.93731876444265, 11.14584433658919, 11.74195715797750, 12.41505200404068, 12.69399907571201
