| L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s − 2·13-s + 2·15-s + 6·17-s + 4·19-s + 4·23-s − 25-s + 27-s + 6·29-s − 4·31-s − 33-s − 37-s − 2·39-s − 6·41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s + 6·51-s + 10·53-s − 2·55-s + 4·57-s − 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s − 0.164·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.840·51-s + 1.37·53-s − 0.269·55-s + 0.529·57-s − 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.573507174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.573507174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.97190920832201, −13.66074071306596, −13.13048650744637, −12.49456206756528, −12.21992823129322, −11.58411446144676, −10.94878404123667, −10.28417121146306, −9.914153918575415, −9.655684546873867, −8.969615028693859, −8.525299350680445, −7.880969013242844, −7.301205851238889, −7.062227385467441, −6.154648605663387, −5.676366681317679, −5.165064115807202, −4.703893931741530, −3.787030312219613, −3.232522190789859, −2.716116446372525, −2.081834033948639, −1.363324183021075, −0.7005306073302050,
0.7005306073302050, 1.363324183021075, 2.081834033948639, 2.716116446372525, 3.232522190789859, 3.787030312219613, 4.703893931741530, 5.165064115807202, 5.676366681317679, 6.154648605663387, 7.062227385467441, 7.301205851238889, 7.880969013242844, 8.525299350680445, 8.969615028693859, 9.655684546873867, 9.914153918575415, 10.28417121146306, 10.94878404123667, 11.58411446144676, 12.21992823129322, 12.49456206756528, 13.13048650744637, 13.66074071306596, 13.97190920832201
