| L(s) = 1 | + 2·5-s − 4·7-s + 4·11-s + 2·13-s + 6·17-s + 4·19-s − 25-s + 2·29-s + 4·31-s − 8·35-s + 2·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s + 10·53-s + 8·55-s − 4·59-s − 6·61-s + 4·65-s − 4·67-s + 16·71-s − 6·73-s − 16·77-s + 4·79-s + 12·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s + 1.07·55-s − 0.520·59-s − 0.768·61-s + 0.496·65-s − 0.488·67-s + 1.89·71-s − 0.702·73-s − 1.82·77-s + 0.450·79-s + 1.31·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.634752516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634752516\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−10.45197268721137434518942050384, −9.629712905768100889040064075295, −9.406971771311537534006960334033, −8.131897207027853144179789700526, −6.83791452023955841983204630481, −6.23593377615022317750995350027, −5.42460705368026111370695370006, −3.81654902962345598726342363363, −2.97479066673588337231673992698, −1.26883614185569776706713350485,
1.26883614185569776706713350485, 2.97479066673588337231673992698, 3.81654902962345598726342363363, 5.42460705368026111370695370006, 6.23593377615022317750995350027, 6.83791452023955841983204630481, 8.131897207027853144179789700526, 9.406971771311537534006960334033, 9.629712905768100889040064075295, 10.45197268721137434518942050384
