| L(s) = 1 | + 3-s − 4·7-s − 2·9-s − 11-s + 2·13-s + 8·17-s − 6·19-s − 4·21-s + 5·23-s − 5·27-s + 4·29-s − 31-s − 33-s − 3·37-s + 2·39-s − 6·41-s − 6·43-s − 12·47-s + 9·49-s + 8·51-s + 6·53-s − 6·57-s − 3·59-s + 8·63-s + 11·67-s + 5·69-s + 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 1.94·17-s − 1.37·19-s − 0.872·21-s + 1.04·23-s − 0.962·27-s + 0.742·29-s − 0.179·31-s − 0.174·33-s − 0.493·37-s + 0.320·39-s − 0.937·41-s − 0.914·43-s − 1.75·47-s + 9/7·49-s + 1.12·51-s + 0.824·53-s − 0.794·57-s − 0.390·59-s + 1.00·63-s + 1.34·67-s + 0.601·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561119316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561119316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−8.039862805326959556490234891581, −6.84682793509602570166454062064, −6.57126349955136101926005728490, −5.68916044406441316193079999795, −5.14953557846758268861061313749, −3.94706008164539649794136872356, −3.18545754625741917783875808503, −3.03433754657910677593949474661, −1.86094429887141455429378765426, −0.56835423137931059611468774622,
0.56835423137931059611468774622, 1.86094429887141455429378765426, 3.03433754657910677593949474661, 3.18545754625741917783875808503, 3.94706008164539649794136872356, 5.14953557846758268861061313749, 5.68916044406441316193079999795, 6.57126349955136101926005728490, 6.84682793509602570166454062064, 8.039862805326959556490234891581
