| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 4·11-s + 4·13-s + 14-s + 16-s + 6·19-s − 20-s − 4·22-s − 2·23-s + 25-s + 4·26-s + 28-s − 8·29-s − 4·31-s + 32-s − 35-s + 4·37-s + 6·38-s − 40-s − 2·41-s − 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 1.48·29-s − 0.718·31-s + 0.176·32-s − 0.169·35-s + 0.657·37-s + 0.973·38-s − 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.279312440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.279312440\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.21740975256249, −12.77846963578144, −12.27322366327320, −11.65613008401289, −11.28210907616911, −11.05182195348889, −10.42757114531822, −9.946177711129732, −9.401573195326636, −8.731420904281392, −8.281211219139292, −7.771333388031518, −7.339630753582614, −7.032759220774257, −6.025608581811371, −5.867545875618260, −5.220409593550760, −4.886510081903263, −4.141380423569804, −3.652205970147803, −3.232536845728272, −2.599233117490508, −1.894516046019873, −1.310867166347895, −0.4571256407261401,
0.4571256407261401, 1.310867166347895, 1.894516046019873, 2.599233117490508, 3.232536845728272, 3.652205970147803, 4.141380423569804, 4.886510081903263, 5.220409593550760, 5.867545875618260, 6.025608581811371, 7.032759220774257, 7.339630753582614, 7.771333388031518, 8.281211219139292, 8.731420904281392, 9.401573195326636, 9.946177711129732, 10.42757114531822, 11.05182195348889, 11.28210907616911, 11.65613008401289, 12.27322366327320, 12.77846963578144, 13.21740975256249
