| L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s + 2·11-s + 13-s − 3·14-s + 16-s − 5·19-s − 20-s − 2·22-s + 25-s − 26-s + 3·28-s − 4·29-s − 31-s − 32-s − 3·35-s + 3·37-s + 5·38-s + 40-s − 6·41-s − 43-s + 2·44-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.277·13-s − 0.801·14-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s − 0.196·26-s + 0.566·28-s − 0.742·29-s − 0.179·31-s − 0.176·32-s − 0.507·35-s + 0.493·37-s + 0.811·38-s + 0.158·40-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−15.45595323413041, −15.15079256744326, −14.59926772987299, −14.21425656638858, −13.40405567552896, −12.87428690676333, −12.11936930124055, −11.73419401118322, −11.18860589495727, −10.77589268670988, −10.24823745102484, −9.461092415653061, −8.883923066172201, −8.471941104087525, −7.893624983762654, −7.450086635202152, −6.690697989287635, −6.213584881185054, −5.385611948607119, −4.726262179851295, −4.045018539686594, −3.466278144075519, −2.419578974998903, −1.777255431675618, −1.077564661912271, 0,
1.077564661912271, 1.777255431675618, 2.419578974998903, 3.466278144075519, 4.045018539686594, 4.726262179851295, 5.385611948607119, 6.213584881185054, 6.690697989287635, 7.450086635202152, 7.893624983762654, 8.471941104087525, 8.883923066172201, 9.461092415653061, 10.24823745102484, 10.77589268670988, 11.18860589495727, 11.73419401118322, 12.11936930124055, 12.87428690676333, 13.40405567552896, 14.21425656638858, 14.59926772987299, 15.15079256744326, 15.45595323413041
