| L(s) = 1 | − 5-s − 7-s − 11-s − 2·13-s − 3·17-s + 5·19-s − 23-s + 25-s − 9·29-s + 4·31-s + 35-s + 2·37-s + 43-s + 49-s + 9·53-s + 55-s + 3·59-s + 7·61-s + 2·65-s − 14·67-s − 12·73-s + 77-s − 14·79-s − 9·83-s + 3·85-s + 11·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.14·19-s − 0.208·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.152·43-s + 1/7·49-s + 1.23·53-s + 0.134·55-s + 0.390·59-s + 0.896·61-s + 0.248·65-s − 1.71·67-s − 1.40·73-s + 0.113·77-s − 1.57·79-s − 0.987·83-s + 0.325·85-s + 1.16·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9508589112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9508589112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.05364169659017, −12.47514419087670, −11.97788528442341, −11.58769357269175, −11.21472048237841, −10.66053772907863, −9.974775021678714, −9.876806426401122, −9.167679173953223, −8.763548422684498, −8.280795378146010, −7.580213610751818, −7.277741422902757, −6.986699329585030, −6.142249692072113, −5.753703303588229, −5.256638776846257, −4.599048121855209, −4.181831120372570, −3.583709912244905, −3.001341574008754, −2.518328079316557, −1.859082270901627, −1.078193469926440, −0.2927614199321908,
0.2927614199321908, 1.078193469926440, 1.859082270901627, 2.518328079316557, 3.001341574008754, 3.583709912244905, 4.181831120372570, 4.599048121855209, 5.256638776846257, 5.753703303588229, 6.142249692072113, 6.986699329585030, 7.277741422902757, 7.580213610751818, 8.280795378146010, 8.763548422684498, 9.167679173953223, 9.876806426401122, 9.974775021678714, 10.66053772907863, 11.21472048237841, 11.58769357269175, 11.97788528442341, 12.47514419087670, 13.05364169659017
