| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 15-s − 4·17-s + 4·19-s + 6·23-s + 25-s + 27-s + 4·29-s + 10·31-s − 4·33-s − 4·37-s + 2·41-s − 12·43-s − 45-s − 7·49-s − 4·51-s − 2·53-s + 4·55-s + 4·57-s − 4·59-s − 10·61-s − 10·67-s + 6·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.258·15-s − 0.970·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.79·31-s − 0.696·33-s − 0.657·37-s + 0.312·41-s − 1.82·43-s − 0.149·45-s − 49-s − 0.560·51-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 1.22·67-s + 0.722·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064951211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064951211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.99388487681240, −14.12715053978063, −13.63056749475628, −13.40045502002201, −12.73168846518159, −12.22965608165373, −11.64852231394635, −11.05056824349261, −10.59800912105858, −9.993063247021462, −9.506347559929221, −8.809763568819190, −8.333270802572870, −7.904199084199348, −7.330011896151109, −6.708090957603105, −6.244592332061938, −5.169149240268655, −4.877390115882612, −4.330888350794202, −3.217519270229268, −3.097416383203662, −2.321962494538675, −1.435473900542590, −0.5109703614484098,
0.5109703614484098, 1.435473900542590, 2.321962494538675, 3.097416383203662, 3.217519270229268, 4.330888350794202, 4.877390115882612, 5.169149240268655, 6.244592332061938, 6.708090957603105, 7.330011896151109, 7.904199084199348, 8.333270802572870, 8.809763568819190, 9.506347559929221, 9.993063247021462, 10.59800912105858, 11.05056824349261, 11.64852231394635, 12.22965608165373, 12.73168846518159, 13.40045502002201, 13.63056749475628, 14.12715053978063, 14.99388487681240
