| L(s) = 1 | − 5-s − 4·7-s + 4·11-s − 2·13-s + 17-s + 8·19-s + 25-s − 10·29-s + 10·31-s + 4·35-s + 8·37-s + 2·41-s + 4·43-s + 6·47-s + 9·49-s + 6·53-s − 4·55-s − 8·59-s + 2·61-s + 2·65-s − 4·67-s + 12·73-s − 16·77-s − 10·79-s + 6·83-s − 85-s − 8·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.20·11-s − 0.554·13-s + 0.242·17-s + 1.83·19-s + 1/5·25-s − 1.85·29-s + 1.79·31-s + 0.676·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 1.04·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s + 1.40·73-s − 1.82·77-s − 1.12·79-s + 0.658·83-s − 0.108·85-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.978061696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978061696\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.52115682264651, −13.98568690778526, −13.55845708737691, −12.98555549756376, −12.39356925877485, −12.04546758809133, −11.55274254176174, −11.07154647507484, −10.22193093796097, −9.748953191996862, −9.333092225754860, −9.109171502049172, −8.130844607160345, −7.587958355542106, −7.067417232982875, −6.665894767824934, −5.814552226301369, −5.678112264428803, −4.570966790412757, −4.111691137270914, −3.382621789103544, −3.055461793564076, −2.261698000830541, −1.130726665381398, −0.5761331928344952,
0.5761331928344952, 1.130726665381398, 2.261698000830541, 3.055461793564076, 3.382621789103544, 4.111691137270914, 4.570966790412757, 5.678112264428803, 5.814552226301369, 6.665894767824934, 7.067417232982875, 7.587958355542106, 8.130844607160345, 9.109171502049172, 9.333092225754860, 9.748953191996862, 10.22193093796097, 11.07154647507484, 11.55274254176174, 12.04546758809133, 12.39356925877485, 12.98555549756376, 13.55845708737691, 13.98568690778526, 14.52115682264651
