| L(s) = 1 | − 5-s − 4·11-s + 13-s − 2·17-s − 19-s + 7·23-s − 4·25-s − 5·29-s + 9·31-s + 2·37-s + 2·41-s − 43-s + 9·47-s + 3·53-s + 4·55-s + 14·61-s − 65-s − 10·67-s + 14·71-s − 3·73-s + 5·79-s − 5·83-s + 2·85-s − 9·89-s + 95-s + 97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.229·19-s + 1.45·23-s − 4/5·25-s − 0.928·29-s + 1.61·31-s + 0.328·37-s + 0.312·41-s − 0.152·43-s + 1.31·47-s + 0.412·53-s + 0.539·55-s + 1.79·61-s − 0.124·65-s − 1.22·67-s + 1.66·71-s − 0.351·73-s + 0.562·79-s − 0.548·83-s + 0.216·85-s − 0.953·89-s + 0.102·95-s + 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112137863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112137863\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.44956310897357, −12.13191332225406, −11.46991735083819, −11.17710401791897, −10.77333581253569, −10.30689173899956, −9.818582382391292, −9.339805506873972, −8.800812173278768, −8.339684966711782, −7.988436508524855, −7.463174813393492, −7.036249595301315, −6.576423216913324, −5.915703226362817, −5.486933582927248, −5.050021941138777, −4.386438593254099, −4.098335470501718, −3.390352042668702, −2.844387322921183, −2.403335164239311, −1.815528951782208, −0.8939548236581114, −0.4587949687220002,
0.4587949687220002, 0.8939548236581114, 1.815528951782208, 2.403335164239311, 2.844387322921183, 3.390352042668702, 4.098335470501718, 4.386438593254099, 5.050021941138777, 5.486933582927248, 5.915703226362817, 6.576423216913324, 7.036249595301315, 7.463174813393492, 7.988436508524855, 8.339684966711782, 8.800812173278768, 9.339805506873972, 9.818582382391292, 10.30689173899956, 10.77333581253569, 11.17710401791897, 11.46991735083819, 12.13191332225406, 12.44956310897357
