| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 4·11-s − 14-s + 16-s + 6·17-s − 3·18-s − 4·22-s − 8·23-s − 28-s − 10·29-s + 8·31-s + 32-s + 6·34-s − 3·36-s + 6·37-s + 6·41-s − 4·43-s − 4·44-s − 8·46-s − 8·47-s + 49-s − 6·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.852·22-s − 1.66·23-s − 0.188·28-s − 1.85·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.603·44-s − 1.17·46-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783291806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783291806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.26214594354328, −13.88934415560351, −13.29499445280188, −12.88587064388501, −12.29592614662275, −11.95505326003017, −11.24498664038361, −10.98701727252873, −10.21967894830022, −9.737539191387781, −9.438027520292348, −8.337044297752280, −7.940946966078279, −7.793376738366536, −6.872290700582710, −6.194114196922590, −5.781596372061041, −5.373186185145382, −4.793281172855825, −3.931013774272122, −3.482206691304111, −2.767518331497988, −2.397979190202953, −1.481717613791522, −0.3875698625679804,
0.3875698625679804, 1.481717613791522, 2.397979190202953, 2.767518331497988, 3.482206691304111, 3.931013774272122, 4.793281172855825, 5.373186185145382, 5.781596372061041, 6.194114196922590, 6.872290700582710, 7.793376738366536, 7.940946966078279, 8.337044297752280, 9.438027520292348, 9.737539191387781, 10.21967894830022, 10.98701727252873, 11.24498664038361, 11.95505326003017, 12.29592614662275, 12.88587064388501, 13.29499445280188, 13.88934415560351, 14.26214594354328
