| L(s) = 1 | + 5-s − 7-s − 3·11-s − 2·13-s + 2·17-s + 3·23-s − 4·25-s + 4·29-s − 2·31-s − 35-s − 3·37-s − 3·41-s + 4·43-s − 47-s − 6·49-s − 5·53-s − 3·55-s + 6·59-s − 12·61-s − 2·65-s − 14·67-s − 6·71-s + 9·73-s + 3·77-s − 6·79-s − 3·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.904·11-s − 0.554·13-s + 0.485·17-s + 0.625·23-s − 4/5·25-s + 0.742·29-s − 0.359·31-s − 0.169·35-s − 0.493·37-s − 0.468·41-s + 0.609·43-s − 0.145·47-s − 6/7·49-s − 0.686·53-s − 0.404·55-s + 0.781·59-s − 1.53·61-s − 0.248·65-s − 1.71·67-s − 0.712·71-s + 1.05·73-s + 0.341·77-s − 0.675·79-s − 0.329·83-s + 0.216·85-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
| \(L(1)\) |
\(\approx\) |
\(0.6022120937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6022120937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 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 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.99842456598592, −12.52578485633419, −11.99916628643901, −11.69848025947749, −10.84391827909598, −10.68350910368865, −10.08795416724351, −9.683400204535931, −9.343028341369653, −8.703460663984744, −8.222332644416353, −7.644768797813879, −7.353489229581498, −6.675612374972986, −6.255065467281806, −5.642624447523623, −5.262440244818488, −4.771516691737882, −4.165634716183321, −3.499402667625614, −2.806865582708181, −2.660554447903932, −1.731500393705268, −1.281643458045978, −0.2087122838252009,
0.2087122838252009, 1.281643458045978, 1.731500393705268, 2.660554447903932, 2.806865582708181, 3.499402667625614, 4.165634716183321, 4.771516691737882, 5.262440244818488, 5.642624447523623, 6.255065467281806, 6.675612374972986, 7.353489229581498, 7.644768797813879, 8.222332644416353, 8.703460663984744, 9.343028341369653, 9.683400204535931, 10.08795416724351, 10.68350910368865, 10.84391827909598, 11.69848025947749, 11.99916628643901, 12.52578485633419, 12.99842456598592
