| L(s) = 1 | + 5-s − 4·7-s − 11-s + 13-s − 6·17-s + 4·19-s − 4·23-s + 25-s − 6·29-s − 4·35-s + 2·37-s − 6·41-s − 4·43-s − 8·47-s + 9·49-s − 2·53-s − 55-s − 12·59-s − 2·61-s + 65-s − 8·67-s + 6·73-s + 4·77-s − 8·79-s − 12·83-s − 6·85-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.676·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s − 0.134·55-s − 1.56·59-s − 0.256·61-s + 0.124·65-s − 0.977·67-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-s − 0.650·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.97017181690153, −13.63685037250970, −13.25810882975823, −12.86588350932525, −12.39144081683320, −11.78799790438823, −11.22264119335868, −10.81521098215792, −10.11247393638907, −9.764521873426440, −9.405681979122673, −8.866426471353936, −8.330696354295214, −7.674563006271787, −7.047474508905620, −6.635714383786839, −6.145457719496595, −5.704450590966093, −5.082188243679694, −4.376340372868033, −3.806253136715024, −3.125911773425120, −2.786736804345249, −1.942310927617893, −1.365896719983630, 0, 0,
1.365896719983630, 1.942310927617893, 2.786736804345249, 3.125911773425120, 3.806253136715024, 4.376340372868033, 5.082188243679694, 5.704450590966093, 6.145457719496595, 6.635714383786839, 7.047474508905620, 7.674563006271787, 8.330696354295214, 8.866426471353936, 9.405681979122673, 9.764521873426440, 10.11247393638907, 10.81521098215792, 11.22264119335868, 11.78799790438823, 12.39144081683320, 12.86588350932525, 13.25810882975823, 13.63685037250970, 13.97017181690153
