| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 2·11-s + 2·14-s + 16-s + 17-s − 4·19-s − 20-s + 2·22-s − 8·23-s + 25-s + 2·28-s − 6·29-s − 4·31-s + 32-s + 34-s − 2·35-s − 4·38-s − 40-s + 4·41-s − 4·43-s + 2·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.648·38-s − 0.158·40-s + 0.624·41-s − 0.609·43-s + 0.301·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.92384806945509, −12.69173935260155, −11.96462760532235, −11.82913363390679, −11.32583709486945, −10.86769653021829, −10.43341682929026, −9.920612869997513, −9.282386611894973, −8.846332981204432, −8.232596029880838, −7.857589535462934, −7.474877183188841, −6.804486099285684, −6.445045306516203, −5.751499226285299, −5.472123474324318, −4.833680026229683, −4.153796027424138, −3.991463259672771, −3.508166757044109, −2.684894258427809, −2.009369575526654, −1.741063774833091, −0.8555197980103918, 0,
0.8555197980103918, 1.741063774833091, 2.009369575526654, 2.684894258427809, 3.508166757044109, 3.991463259672771, 4.153796027424138, 4.833680026229683, 5.472123474324318, 5.751499226285299, 6.445045306516203, 6.804486099285684, 7.474877183188841, 7.857589535462934, 8.232596029880838, 8.846332981204432, 9.282386611894973, 9.920612869997513, 10.43341682929026, 10.86769653021829, 11.32583709486945, 11.82913363390679, 11.96462760532235, 12.69173935260155, 12.92384806945509
