| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 4·5-s − 2·6-s − 2·7-s + 9-s + 8·10-s + 5·11-s − 2·12-s − 4·14-s − 4·15-s − 4·16-s − 7·17-s + 2·18-s + 8·20-s + 2·21-s + 10·22-s + 6·23-s + 11·25-s − 27-s − 4·28-s − 29-s − 8·30-s + 5·31-s − 8·32-s − 5·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s + 2.52·10-s + 1.50·11-s − 0.577·12-s − 1.06·14-s − 1.03·15-s − 16-s − 1.69·17-s + 0.471·18-s + 1.78·20-s + 0.436·21-s + 2.13·22-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.755·28-s − 0.185·29-s − 1.46·30-s + 0.898·31-s − 1.41·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.794483224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.794483224\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.39512225456464, −14.87529318247712, −14.48445170973089, −13.71840310348511, −13.43338598269684, −13.06296713984250, −12.60252747832723, −11.90948547944525, −11.29075787537992, −10.92242403770262, −9.992667271718948, −9.565351390335921, −9.078746279651203, −8.595284200823825, −7.034300118974245, −6.712612062561339, −6.265265485855801, −5.952683024507625, −5.175770488395753, −4.616309389662362, −4.063187935124134, −3.149439862824230, −2.507926332188340, −1.799519961345142, −0.8183043303397443,
0.8183043303397443, 1.799519961345142, 2.507926332188340, 3.149439862824230, 4.063187935124134, 4.616309389662362, 5.175770488395753, 5.952683024507625, 6.265265485855801, 6.712612062561339, 7.034300118974245, 8.595284200823825, 9.078746279651203, 9.565351390335921, 9.992667271718948, 10.92242403770262, 11.29075787537992, 11.90948547944525, 12.60252747832723, 13.06296713984250, 13.43338598269684, 13.71840310348511, 14.48445170973089, 14.87529318247712, 15.39512225456464
