| L(s) = 1 | + 2·3-s + 7-s + 9-s − 11-s + 4·13-s − 6·19-s + 2·21-s − 3·23-s − 4·27-s + 3·29-s − 2·33-s + 9·37-s + 8·39-s + 2·41-s + 9·43-s + 6·47-s + 49-s + 6·53-s − 12·57-s − 8·59-s + 10·61-s + 63-s − 67-s − 6·69-s − 7·71-s + 2·73-s − 77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 1.37·19-s + 0.436·21-s − 0.625·23-s − 0.769·27-s + 0.557·29-s − 0.348·33-s + 1.47·37-s + 1.28·39-s + 0.312·41-s + 1.37·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.58·57-s − 1.04·59-s + 1.28·61-s + 0.125·63-s − 0.122·67-s − 0.722·69-s − 0.830·71-s + 0.234·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.397959847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.397959847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.41047619862593, −15.68345841695281, −15.36009903600009, −14.61316278591286, −14.30151524042233, −13.67436794204668, −13.17216346580783, −12.66118619791704, −11.87974898548965, −11.14544239348860, −10.71199398274419, −10.01462335677472, −9.267651832936172, −8.685916921504128, −8.319129483689240, −7.754006248204738, −7.075601265321753, −6.044968413487682, −5.811526907672710, −4.516702181200856, −4.119920929493603, −3.313196895933596, −2.493616654356396, −1.960480438819704, −0.7998683516467595,
0.7998683516467595, 1.960480438819704, 2.493616654356396, 3.313196895933596, 4.119920929493603, 4.516702181200856, 5.811526907672710, 6.044968413487682, 7.075601265321753, 7.754006248204738, 8.319129483689240, 8.685916921504128, 9.267651832936172, 10.01462335677472, 10.71199398274419, 11.14544239348860, 11.87974898548965, 12.66118619791704, 13.17216346580783, 13.67436794204668, 14.30151524042233, 14.61316278591286, 15.36009903600009, 15.68345841695281, 16.41047619862593
