| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 13-s − 4·19-s + 2·21-s + 4·23-s − 27-s − 2·29-s + 4·33-s − 6·37-s + 39-s − 2·41-s − 12·43-s + 2·47-s − 3·49-s + 2·53-s + 4·57-s + 10·61-s − 2·63-s − 4·67-s − 4·69-s − 6·73-s + 8·77-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.917·19-s + 0.436·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.160·39-s − 0.312·41-s − 1.82·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s + 0.529·57-s + 1.28·61-s − 0.251·63-s − 0.488·67-s − 0.481·69-s − 0.702·73-s + 0.911·77-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.85100076258564, −13.28729835127552, −13.06710564343506, −12.64618028031736, −12.13948541947115, −11.56530924322405, −11.11557946530156, −10.55895359077104, −10.09849313520898, −9.931844158471188, −9.080528174097851, −8.679674422950874, −8.116492757660045, −7.511206980393617, −6.973325752819708, −6.603097011940784, −6.030264900162981, −5.389733679015297, −5.044636436065312, −4.502032017103210, −3.709637281950633, −3.225728331442012, −2.564122443611841, −1.970058215573787, −1.145947263107574, 0, 0,
1.145947263107574, 1.970058215573787, 2.564122443611841, 3.225728331442012, 3.709637281950633, 4.502032017103210, 5.044636436065312, 5.389733679015297, 6.030264900162981, 6.603097011940784, 6.973325752819708, 7.511206980393617, 8.116492757660045, 8.679674422950874, 9.080528174097851, 9.931844158471188, 10.09849313520898, 10.55895359077104, 11.11557946530156, 11.56530924322405, 12.13948541947115, 12.64618028031736, 13.06710564343506, 13.28729835127552, 13.85100076258564
