| L(s) = 1 | − 5-s − 7-s − 11-s + 5·17-s − 3·19-s − 7·23-s − 4·25-s + 5·29-s + 4·31-s + 35-s − 3·37-s + 5·43-s + 49-s + 6·53-s + 55-s − 4·59-s + 5·61-s + 6·67-s − 8·71-s − 73-s + 77-s − 10·79-s − 6·83-s − 5·85-s + 14·89-s + 3·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.21·17-s − 0.688·19-s − 1.45·23-s − 4/5·25-s + 0.928·29-s + 0.718·31-s + 0.169·35-s − 0.493·37-s + 0.762·43-s + 1/7·49-s + 0.824·53-s + 0.134·55-s − 0.520·59-s + 0.640·61-s + 0.733·67-s − 0.949·71-s − 0.117·73-s + 0.113·77-s − 1.12·79-s − 0.658·83-s − 0.542·85-s + 1.48·89-s + 0.307·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.538254879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538254879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−12.59489528323078, −12.04253680507839, −11.73112218353750, −11.41454412201756, −10.53565587120777, −10.24898186421376, −10.05784186368716, −9.439753825644933, −8.816661835925773, −8.436966066124498, −7.900501001234298, −7.613957699667220, −7.110498013383069, −6.384340466332473, −6.105944444379508, −5.607461571637757, −5.043320794842475, −4.395461210939013, −4.006969082224597, −3.492907087087688, −2.952138338629921, −2.338770431308239, −1.808525843016009, −0.9798140284340118, −0.3691496570405942,
0.3691496570405942, 0.9798140284340118, 1.808525843016009, 2.338770431308239, 2.952138338629921, 3.492907087087688, 4.006969082224597, 4.395461210939013, 5.043320794842475, 5.607461571637757, 6.105944444379508, 6.384340466332473, 7.110498013383069, 7.613957699667220, 7.900501001234298, 8.436966066124498, 8.816661835925773, 9.439753825644933, 10.05784186368716, 10.24898186421376, 10.53565587120777, 11.41454412201756, 11.73112218353750, 12.04253680507839, 12.59489528323078
