| L(s) = 1 | + 3-s − 3·7-s − 2·9-s + 5·11-s + 2·13-s − 3·17-s + 6·19-s − 3·21-s − 8·23-s − 5·27-s − 9·29-s + 9·31-s + 5·33-s − 9·37-s + 2·39-s − 10·41-s + 6·43-s − 6·47-s + 2·49-s − 3·51-s − 2·53-s + 6·57-s − 59-s + 7·61-s + 6·63-s − 8·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s + 1.50·11-s + 0.554·13-s − 0.727·17-s + 1.37·19-s − 0.654·21-s − 1.66·23-s − 0.962·27-s − 1.67·29-s + 1.61·31-s + 0.870·33-s − 1.47·37-s + 0.320·39-s − 1.56·41-s + 0.914·43-s − 0.875·47-s + 2/7·49-s − 0.420·51-s − 0.274·53-s + 0.794·57-s − 0.130·59-s + 0.896·61-s + 0.755·63-s − 0.977·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402408356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402408356\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.59161849227157, −13.22364325114709, −12.24124356999730, −12.19842642471623, −11.52363274020444, −11.25167733755715, −10.49603300248995, −9.849209635971089, −9.495904483823332, −9.235750505606919, −8.575210125530686, −8.219058316963005, −7.632065639176828, −6.883528008978711, −6.533148572130683, −6.127573221124098, −5.536250912492188, −4.946115047389484, −3.927060031694201, −3.756636558045070, −3.313498741949341, −2.634522034050529, −1.918174690837218, −1.336670990974002, −0.3369945234773998,
0.3369945234773998, 1.336670990974002, 1.918174690837218, 2.634522034050529, 3.313498741949341, 3.756636558045070, 3.927060031694201, 4.946115047389484, 5.536250912492188, 6.127573221124098, 6.533148572130683, 6.883528008978711, 7.632065639176828, 8.219058316963005, 8.575210125530686, 9.235750505606919, 9.495904483823332, 9.849209635971089, 10.49603300248995, 11.25167733755715, 11.52363274020444, 12.19842642471623, 12.24124356999730, 13.22364325114709, 13.59161849227157
