| L(s) = 1 | − 2·4-s − 5-s − 3·11-s + 4·16-s − 3·17-s − 4·19-s + 2·20-s + 9·23-s + 25-s + 6·29-s + 2·31-s + 37-s + 3·41-s + 2·43-s + 6·44-s + 6·47-s − 9·53-s + 3·55-s + 12·59-s − 5·61-s − 8·64-s + 4·67-s + 6·68-s + 9·71-s + 14·73-s + 8·76-s − 7·79-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.904·11-s + 16-s − 0.727·17-s − 0.917·19-s + 0.447·20-s + 1.87·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s + 0.164·37-s + 0.468·41-s + 0.304·43-s + 0.904·44-s + 0.875·47-s − 1.23·53-s + 0.404·55-s + 1.56·59-s − 0.640·61-s − 64-s + 0.488·67-s + 0.727·68-s + 1.06·71-s + 1.63·73-s + 0.917·76-s − 0.787·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754077098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754077098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.61226840584288, −12.31801620784187, −11.41543309472443, −11.15182290674396, −10.63740935265259, −10.35989260872358, −9.641484354524316, −9.361191911236166, −8.731932891438881, −8.456584247991148, −8.049649687124698, −7.575462511973704, −6.826581323881293, −6.693192405622935, −5.883975022058511, −5.343617569615586, −4.971714195224127, −4.396921166960785, −4.200544359954719, −3.444410319478481, −2.858058168973274, −2.512182126301672, −1.653220430104753, −0.7722765388614652, −0.4945450700436527,
0.4945450700436527, 0.7722765388614652, 1.653220430104753, 2.512182126301672, 2.858058168973274, 3.444410319478481, 4.200544359954719, 4.396921166960785, 4.971714195224127, 5.343617569615586, 5.883975022058511, 6.693192405622935, 6.826581323881293, 7.575462511973704, 8.049649687124698, 8.456584247991148, 8.731932891438881, 9.361191911236166, 9.641484354524316, 10.35989260872358, 10.63740935265259, 11.15182290674396, 11.41543309472443, 12.31801620784187, 12.61226840584288
