| L(s) = 1 | + 3-s + 7-s + 9-s − 5·11-s − 2·13-s − 17-s − 4·19-s + 21-s − 7·23-s + 27-s + 6·29-s − 5·31-s − 5·33-s − 2·39-s − 10·41-s + 12·43-s + 7·47-s + 49-s − 51-s − 4·53-s − 4·57-s − 9·59-s − 3·61-s + 63-s + 11·67-s − 7·69-s + 7·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.218·21-s − 1.45·23-s + 0.192·27-s + 1.11·29-s − 0.898·31-s − 0.870·33-s − 0.320·39-s − 1.56·41-s + 1.82·43-s + 1.02·47-s + 1/7·49-s − 0.140·51-s − 0.549·53-s − 0.529·57-s − 1.17·59-s − 0.384·61-s + 0.125·63-s + 1.34·67-s − 0.842·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484589761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484589761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−14.97446949745895, −14.28277268873937, −13.96109188509863, −13.44920283854109, −12.72937915084126, −12.48646513445201, −11.87417620661943, −11.11103685410270, −10.48016419386002, −10.33236190917609, −9.585334170490226, −8.979144769244599, −8.331805494800278, −7.985917885068722, −7.470579460421849, −6.866159128096843, −6.074999323125556, −5.526566675228752, −4.790823547844959, −4.353044299624897, −3.608729952009694, −2.746841147729526, −2.305927543447686, −1.677980269644328, −0.4106876441717064,
0.4106876441717064, 1.677980269644328, 2.305927543447686, 2.746841147729526, 3.608729952009694, 4.353044299624897, 4.790823547844959, 5.526566675228752, 6.074999323125556, 6.866159128096843, 7.470579460421849, 7.985917885068722, 8.331805494800278, 8.979144769244599, 9.585334170490226, 10.33236190917609, 10.48016419386002, 11.11103685410270, 11.87417620661943, 12.48646513445201, 12.72937915084126, 13.44920283854109, 13.96109188509863, 14.28277268873937, 14.97446949745895
