| L(s) = 1 | − 4·7-s + 4·11-s + 6·13-s − 8·17-s + 4·19-s − 8·23-s − 5·25-s − 2·29-s + 8·31-s − 10·37-s + 12·41-s − 4·43-s − 12·47-s + 9·49-s − 4·53-s − 6·59-s − 6·61-s − 4·67-s + 14·71-s − 6·73-s − 16·77-s − 8·79-s + 16·83-s + 12·89-s − 24·91-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s + 1.66·13-s − 1.94·17-s + 0.917·19-s − 1.66·23-s − 25-s − 0.371·29-s + 1.43·31-s − 1.64·37-s + 1.87·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.549·53-s − 0.781·59-s − 0.768·61-s − 0.488·67-s + 1.66·71-s − 0.702·73-s − 1.82·77-s − 0.900·79-s + 1.75·83-s + 1.27·89-s − 2.51·91-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.29392598429034, −12.77664587479296, −12.15420190458782, −11.78911224865242, −11.41235252132232, −10.81725623730289, −10.41332018685155, −9.729187934461350, −9.415334486365675, −9.120422194141010, −8.486765858438881, −8.104559575910694, −7.447573782086117, −6.668433958957970, −6.465215384913700, −6.182597579274013, −5.728162828430397, −4.852428590564967, −4.099098083443017, −3.956872864260240, −3.339616169798298, −2.903044395133743, −1.948697497400434, −1.611754588059993, −0.6841590888119379, 0,
0.6841590888119379, 1.611754588059993, 1.948697497400434, 2.903044395133743, 3.339616169798298, 3.956872864260240, 4.099098083443017, 4.852428590564967, 5.728162828430397, 6.182597579274013, 6.465215384913700, 6.668433958957970, 7.447573782086117, 8.104559575910694, 8.486765858438881, 9.120422194141010, 9.415334486365675, 9.729187934461350, 10.41332018685155, 10.81725623730289, 11.41235252132232, 11.78911224865242, 12.15420190458782, 12.77664587479296, 13.29392598429034
