| L(s) = 1 | + 2·5-s + 4·7-s − 2·13-s + 2·17-s + 4·23-s − 25-s + 29-s + 6·31-s + 8·35-s + 4·37-s + 2·41-s − 4·43-s − 8·47-s + 9·49-s + 14·53-s − 6·59-s + 8·61-s − 4·65-s + 12·67-s − 16·71-s − 2·73-s − 6·79-s + 2·83-s + 4·85-s + 14·89-s − 8·91-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.485·17-s + 0.834·23-s − 1/5·25-s + 0.185·29-s + 1.07·31-s + 1.35·35-s + 0.657·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.92·53-s − 0.781·59-s + 1.02·61-s − 0.496·65-s + 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.675·79-s + 0.219·83-s + 0.433·85-s + 1.48·89-s − 0.838·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.621658198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.621658198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.90254191410515, −15.15753448017208, −14.61356274063992, −14.48092608631784, −13.56256767365519, −13.42190652683864, −12.55125484176368, −11.91365350906675, −11.45166119820668, −10.92445780152007, −10.07219883114387, −9.922302187724429, −9.039209076549156, −8.467540035444928, −7.914465245120552, −7.315265025610809, −6.612172463693736, −5.842079289845969, −5.265555198989153, −4.775212694853968, −4.114962133117372, −3.031259233477688, −2.309252789197920, −1.630269146458621, −0.8604497214396277,
0.8604497214396277, 1.630269146458621, 2.309252789197920, 3.031259233477688, 4.114962133117372, 4.775212694853968, 5.265555198989153, 5.842079289845969, 6.612172463693736, 7.315265025610809, 7.914465245120552, 8.467540035444928, 9.039209076549156, 9.922302187724429, 10.07219883114387, 10.92445780152007, 11.45166119820668, 11.91365350906675, 12.55125484176368, 13.42190652683864, 13.56256767365519, 14.48092608631784, 14.61356274063992, 15.15753448017208, 15.90254191410515
