| L(s) = 1 | − 2·5-s − 2·7-s − 2·13-s + 2·17-s − 25-s − 2·29-s − 4·31-s + 4·35-s + 6·37-s − 6·41-s − 4·43-s + 4·47-s − 3·49-s − 2·53-s + 6·59-s + 6·61-s + 4·65-s + 8·67-s − 6·71-s − 6·73-s + 12·79-s − 12·83-s − 4·85-s − 14·89-s + 4·91-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.554·13-s + 0.485·17-s − 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.274·53-s + 0.781·59-s + 0.768·61-s + 0.496·65-s + 0.977·67-s − 0.712·71-s − 0.702·73-s + 1.35·79-s − 1.31·83-s − 0.433·85-s − 1.48·89-s + 0.419·91-s − 1.42·97-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 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.09406311351424, −12.58105073173325, −12.38315395220111, −11.67003148157234, −11.41167958051342, −11.01009515743735, −10.18790239808945, −9.929492390982553, −9.581330086358676, −8.835496604528641, −8.539727402408508, −7.802252128207522, −7.599936218085216, −7.005561218801835, −6.591352783667971, −5.987047987568612, −5.420787433838319, −4.988250871314768, −4.233146418459838, −3.875010237642309, −3.330697654044085, −2.849901716004266, −2.170943460131270, −1.459662958925567, −0.5899495978004991, 0,
0.5899495978004991, 1.459662958925567, 2.170943460131270, 2.849901716004266, 3.330697654044085, 3.875010237642309, 4.233146418459838, 4.988250871314768, 5.420787433838319, 5.987047987568612, 6.591352783667971, 7.005561218801835, 7.599936218085216, 7.802252128207522, 8.539727402408508, 8.835496604528641, 9.581330086358676, 9.929492390982553, 10.18790239808945, 11.01009515743735, 11.41167958051342, 11.67003148157234, 12.38315395220111, 12.58105073173325, 13.09406311351424
