| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 11-s + 2·13-s + 15-s + 2·21-s + 7·23-s − 4·25-s + 27-s − 9·31-s + 33-s + 2·35-s + 10·37-s + 2·39-s + 4·43-s + 45-s − 3·49-s − 53-s + 55-s + 59-s − 12·61-s + 2·63-s + 2·65-s − 12·67-s + 7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.436·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.61·31-s + 0.174·33-s + 0.338·35-s + 1.64·37-s + 0.320·39-s + 0.609·43-s + 0.149·45-s − 3/7·49-s − 0.137·53-s + 0.134·55-s + 0.130·59-s − 1.53·61-s + 0.251·63-s + 0.248·65-s − 1.46·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.31977417108568, −12.86763943030680, −12.30935312153633, −11.77450680180310, −11.24732150309848, −10.79634610471806, −10.62489341475473, −9.646066012698324, −9.472119754961216, −9.015078768733383, −8.532747939344664, −8.016101405210268, −7.451721392951964, −7.256296140521138, −6.362061227744952, −6.104115092889764, −5.395408519013808, −5.007063852814100, −4.267169324806613, −3.984003947151024, −3.202074214808999, −2.759288608657813, −2.074665304827081, −1.460661064930232, −1.121416583371151, 0,
1.121416583371151, 1.460661064930232, 2.074665304827081, 2.759288608657813, 3.202074214808999, 3.984003947151024, 4.267169324806613, 5.007063852814100, 5.395408519013808, 6.104115092889764, 6.362061227744952, 7.256296140521138, 7.451721392951964, 8.016101405210268, 8.532747939344664, 9.015078768733383, 9.472119754961216, 9.646066012698324, 10.62489341475473, 10.79634610471806, 11.24732150309848, 11.77450680180310, 12.30935312153633, 12.86763943030680, 13.31977417108568
