| L(s) = 1 | + 5-s − 2·7-s + 4·13-s − 2·17-s + 2·19-s + 4·23-s + 25-s − 6·29-s − 2·35-s + 10·37-s − 2·41-s + 2·43-s − 3·49-s − 6·53-s + 4·61-s + 4·65-s − 4·67-s + 2·79-s − 2·85-s − 10·89-s − 8·91-s + 2·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.10·13-s − 0.485·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.338·35-s + 1.64·37-s − 0.312·41-s + 0.304·43-s − 3/7·49-s − 0.824·53-s + 0.512·61-s + 0.496·65-s − 0.488·67-s + 0.225·79-s − 0.216·85-s − 1.05·89-s − 0.838·91-s + 0.205·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.02864421576760, −13.52339288046481, −13.19752752183252, −12.81018991453802, −12.33367342110635, −11.48999025456227, −11.16905470152100, −10.81908851174254, −10.01530167723395, −9.705496450339554, −9.080429560521414, −8.863065327249706, −8.049663664519196, −7.617319554592095, −6.847721274138047, −6.523317964904725, −5.922421347824923, −5.529241621019328, −4.808264288946912, −4.155915197621366, −3.558764491723629, −3.007903314674053, −2.414686351804036, −1.566714143433616, −0.9676199732203847, 0,
0.9676199732203847, 1.566714143433616, 2.414686351804036, 3.007903314674053, 3.558764491723629, 4.155915197621366, 4.808264288946912, 5.529241621019328, 5.922421347824923, 6.523317964904725, 6.847721274138047, 7.617319554592095, 8.049663664519196, 8.863065327249706, 9.080429560521414, 9.705496450339554, 10.01530167723395, 10.81908851174254, 11.16905470152100, 11.48999025456227, 12.33367342110635, 12.81018991453802, 13.19752752183252, 13.52339288046481, 14.02864421576760
