| L(s) = 1 | − 5-s − 4·7-s + 4·13-s + 6·17-s + 7·19-s − 6·23-s − 4·25-s + 5·31-s + 4·35-s + 2·37-s − 10·41-s + 4·43-s − 12·47-s + 9·49-s + 9·53-s + 5·59-s + 6·61-s − 4·65-s − 67-s − 4·71-s − 3·73-s + 5·79-s + 16·83-s − 6·85-s + 89-s − 16·91-s − 7·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.10·13-s + 1.45·17-s + 1.60·19-s − 1.25·23-s − 4/5·25-s + 0.898·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s + 1.23·53-s + 0.650·59-s + 0.768·61-s − 0.496·65-s − 0.122·67-s − 0.474·71-s − 0.351·73-s + 0.562·79-s + 1.75·83-s − 0.650·85-s + 0.105·89-s − 1.67·91-s − 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979469790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979469790\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.06531117475554, −12.32656239439051, −11.93697945056104, −11.76663349028155, −11.20305834465169, −10.40853413435407, −10.00210397503892, −9.813489501699711, −9.359825836762828, −8.610443904759015, −8.241137911780306, −7.689196378288211, −7.334981657116985, −6.622494021438433, −6.258860428595773, −5.762810969545566, −5.369091889861390, −4.649880655827119, −3.810363106580817, −3.542568960540195, −3.272219672971250, −2.571986110122691, −1.733119334767453, −0.9980642583584544, −0.4617388177189107,
0.4617388177189107, 0.9980642583584544, 1.733119334767453, 2.571986110122691, 3.272219672971250, 3.542568960540195, 3.810363106580817, 4.649880655827119, 5.369091889861390, 5.762810969545566, 6.258860428595773, 6.622494021438433, 7.334981657116985, 7.689196378288211, 8.241137911780306, 8.610443904759015, 9.359825836762828, 9.813489501699711, 10.00210397503892, 10.40853413435407, 11.20305834465169, 11.76663349028155, 11.93697945056104, 12.32656239439051, 13.06531117475554
