| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s − 13-s + 4·14-s + 16-s + 6·17-s + 8·19-s − 20-s − 4·23-s + 25-s − 26-s + 4·28-s + 2·29-s + 32-s + 6·34-s − 4·35-s − 10·37-s + 8·38-s − 40-s + 10·41-s − 4·43-s − 4·46-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 1.83·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s + 0.371·29-s + 0.176·32-s + 1.02·34-s − 0.676·35-s − 1.64·37-s + 1.29·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.589·46-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.927166583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.927166583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 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 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.62503646571870, −12.86748207958043, −12.17340769994468, −11.96485946024040, −11.79406912673826, −11.08213250671852, −10.64600935720550, −10.20541773753009, −9.532899165611258, −9.095937815901669, −8.239368457917597, −7.930236128385580, −7.552399168183404, −7.200285199545823, −6.382193871896942, −5.751346673473070, −5.280956597493071, −4.929388143675418, −4.426662427345821, −3.703087878262519, −3.283337010957267, −2.682024232050877, −1.799281103200020, −1.385734448823509, −0.6632962315491734,
0.6632962315491734, 1.385734448823509, 1.799281103200020, 2.682024232050877, 3.283337010957267, 3.703087878262519, 4.426662427345821, 4.929388143675418, 5.280956597493071, 5.751346673473070, 6.382193871896942, 7.200285199545823, 7.552399168183404, 7.930236128385580, 8.239368457917597, 9.095937815901669, 9.532899165611258, 10.20541773753009, 10.64600935720550, 11.08213250671852, 11.79406912673826, 11.96485946024040, 12.17340769994468, 12.86748207958043, 13.62503646571870
