| L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s − 5·13-s + 3·14-s + 16-s + 7·17-s + 7·19-s + 5·26-s − 3·28-s + 7·29-s + 6·31-s − 32-s − 7·34-s + 5·37-s − 7·38-s − 10·41-s + 6·43-s − 10·47-s + 2·49-s − 5·52-s − 12·53-s + 3·56-s − 7·58-s − 12·61-s − 6·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 1.38·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s + 1.60·19-s + 0.980·26-s − 0.566·28-s + 1.29·29-s + 1.07·31-s − 0.176·32-s − 1.20·34-s + 0.821·37-s − 1.13·38-s − 1.56·41-s + 0.914·43-s − 1.45·47-s + 2/7·49-s − 0.693·52-s − 1.64·53-s + 0.400·56-s − 0.919·58-s − 1.53·61-s − 0.762·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.65900538882675, −14.16838378223274, −13.79485349302176, −13.04652261671714, −12.43893127814967, −12.09260657640498, −11.79211488837848, −11.03450858329397, −10.27652160788296, −9.818190030796865, −9.721108581814175, −9.252976210493244, −8.264051557869130, −7.932082296150471, −7.411238132577607, −6.815148041807378, −6.350708399362489, −5.692219585048151, −5.068125642893111, −4.530205169029244, −3.378732982770904, −3.096371732289342, −2.629693131788038, −1.521039285585845, −0.8717915740795053, 0,
0.8717915740795053, 1.521039285585845, 2.629693131788038, 3.096371732289342, 3.378732982770904, 4.530205169029244, 5.068125642893111, 5.692219585048151, 6.350708399362489, 6.815148041807378, 7.411238132577607, 7.932082296150471, 8.264051557869130, 9.252976210493244, 9.721108581814175, 9.818190030796865, 10.27652160788296, 11.03450858329397, 11.79211488837848, 12.09260657640498, 12.43893127814967, 13.04652261671714, 13.79485349302176, 14.16838378223274, 14.65900538882675
