| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s − 11-s + 12-s − 2·15-s + 16-s + 17-s − 18-s + 2·19-s − 2·20-s + 22-s − 6·23-s − 24-s − 25-s + 27-s + 2·29-s + 2·30-s − 32-s − 33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.447·20-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.365·30-s − 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9895474451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9895474451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.55690168319761, −13.86580910153408, −13.52026377140259, −12.81191326439460, −12.25539979552880, −11.72977197875579, −11.48917430013360, −10.72309456218267, −10.09463190731687, −9.888706743850188, −9.180521473666185, −8.549780578690049, −8.192991500830318, −7.648486618139898, −7.396951035192600, −6.558332048464386, −6.119097999217149, −5.239137427014981, −4.687098908484187, −3.805230059206216, −3.526849132282839, −2.739202025787152, −2.070436525315453, −1.331711136586320, −0.3807426097251102,
0.3807426097251102, 1.331711136586320, 2.070436525315453, 2.739202025787152, 3.526849132282839, 3.805230059206216, 4.687098908484187, 5.239137427014981, 6.119097999217149, 6.558332048464386, 7.396951035192600, 7.648486618139898, 8.192991500830318, 8.549780578690049, 9.180521473666185, 9.888706743850188, 10.09463190731687, 10.72309456218267, 11.48917430013360, 11.72977197875579, 12.25539979552880, 12.81191326439460, 13.52026377140259, 13.86580910153408, 14.55690168319761
