| L(s) = 1 | + 2-s + 4-s − 3·5-s − 7-s + 8-s − 3·10-s − 3·11-s − 14-s + 16-s − 17-s − 2·19-s − 3·20-s − 3·22-s + 4·25-s − 28-s − 2·31-s + 32-s − 34-s + 3·35-s + 37-s − 2·38-s − 3·40-s + 5·43-s − 3·44-s + 6·47-s + 49-s + 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.670·20-s − 0.639·22-s + 4/5·25-s − 0.188·28-s − 0.359·31-s + 0.176·32-s − 0.171·34-s + 0.507·35-s + 0.164·37-s − 0.324·38-s − 0.474·40-s + 0.762·43-s − 0.452·44-s + 0.875·47-s + 1/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677029641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677029641\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.57637462743813, −12.10950591241728, −11.67669858836384, −11.22576892141996, −10.85562296672426, −10.41169481266777, −9.957649052770703, −9.288389668815979, −8.792170735879031, −8.293649825914254, −7.687395606753979, −7.661158987507625, −6.845329205169406, −6.651481793560153, −5.932670397881723, −5.361831035040288, −5.072578290636533, −4.211562260094077, −4.123810708497031, −3.564243037199002, −2.950414233446505, −2.504916276570178, −1.940272302641511, −0.9624775425570298, −0.3377803605367080,
0.3377803605367080, 0.9624775425570298, 1.940272302641511, 2.504916276570178, 2.950414233446505, 3.564243037199002, 4.123810708497031, 4.211562260094077, 5.072578290636533, 5.361831035040288, 5.932670397881723, 6.651481793560153, 6.845329205169406, 7.661158987507625, 7.687395606753979, 8.293649825914254, 8.792170735879031, 9.288389668815979, 9.957649052770703, 10.41169481266777, 10.85562296672426, 11.22576892141996, 11.67669858836384, 12.10950591241728, 12.57637462743813
