| L(s) = 1 | + 2·5-s − 2·7-s + 6·11-s + 2·13-s − 17-s − 6·23-s − 25-s + 10·29-s + 2·31-s − 4·35-s + 6·37-s + 6·41-s − 8·43-s − 3·49-s + 10·53-s + 12·55-s + 8·59-s + 14·61-s + 4·65-s + 4·67-s − 2·71-s − 14·73-s − 12·77-s − 10·79-s − 8·83-s − 2·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 1.80·11-s + 0.554·13-s − 0.242·17-s − 1.25·23-s − 1/5·25-s + 1.85·29-s + 0.359·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 3/7·49-s + 1.37·53-s + 1.61·55-s + 1.04·59-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 0.237·71-s − 1.63·73-s − 1.36·77-s − 1.12·79-s − 0.878·83-s − 0.216·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005174125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005174125\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.912053904035992312982696740571, −8.965330077436591257889235047076, −8.331420707616764826583042598870, −6.95556909204739164167709631680, −6.32076764353867371001165550609, −5.84493337287348416539622628292, −4.42426362198009611408546968702, −3.62388232922820390627798777230, −2.37082491510612846557132007360, −1.14755912620769522242228089263,
1.14755912620769522242228089263, 2.37082491510612846557132007360, 3.62388232922820390627798777230, 4.42426362198009611408546968702, 5.84493337287348416539622628292, 6.32076764353867371001165550609, 6.95556909204739164167709631680, 8.331420707616764826583042598870, 8.965330077436591257889235047076, 9.912053904035992312982696740571
