| L(s) = 1 | + 3-s + 4·7-s + 9-s + 4·11-s + 2·13-s + 6·17-s + 4·19-s + 4·21-s + 8·23-s + 27-s + 6·29-s + 4·31-s + 4·33-s − 2·37-s + 2·39-s − 2·41-s + 4·43-s + 9·49-s + 6·51-s + 10·53-s + 4·57-s − 10·61-s + 4·63-s − 67-s + 8·69-s − 10·73-s + 16·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.66·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 9/7·49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s − 1.28·61-s + 0.503·63-s − 0.122·67-s + 0.963·69-s − 1.17·73-s + 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.493995460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.493995460\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.42726311841696, −12.08371870738074, −11.78667897697876, −11.35057857950995, −10.73467974591219, −10.46897019514103, −9.793720300228265, −9.372649384149462, −8.807632611202706, −8.583862384835095, −8.000909354906902, −7.653509368751405, −7.071944416165116, −6.760882636914336, −5.967680019503004, −5.530989808645341, −4.973417763100980, −4.543764476234493, −4.037807183580410, −3.393107524691512, −3.007167415028304, −2.383809958165284, −1.432224771031528, −1.272490942962727, −0.8569327167465351,
0.8569327167465351, 1.272490942962727, 1.432224771031528, 2.383809958165284, 3.007167415028304, 3.393107524691512, 4.037807183580410, 4.543764476234493, 4.973417763100980, 5.530989808645341, 5.967680019503004, 6.760882636914336, 7.071944416165116, 7.653509368751405, 8.000909354906902, 8.583862384835095, 8.807632611202706, 9.372649384149462, 9.793720300228265, 10.46897019514103, 10.73467974591219, 11.35057857950995, 11.78667897697876, 12.08371870738074, 12.42726311841696
