| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 6·11-s + 13-s + 2·14-s + 16-s + 3·17-s + 5·19-s − 6·22-s − 6·23-s − 26-s − 2·28-s − 31-s − 32-s − 3·34-s − 2·37-s − 5·38-s + 41-s − 8·43-s + 6·44-s + 6·46-s − 12·47-s − 3·49-s + 52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.80·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 1.27·22-s − 1.25·23-s − 0.196·26-s − 0.377·28-s − 0.179·31-s − 0.176·32-s − 0.514·34-s − 0.328·37-s − 0.811·38-s + 0.156·41-s − 1.21·43-s + 0.904·44-s + 0.884·46-s − 1.75·47-s − 3/7·49-s + 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−16.29694658735132, −15.67507757734681, −14.94970375680404, −14.42570885794697, −13.96032876948664, −13.34071995659100, −12.57702017018560, −12.01401407528031, −11.63663337409932, −11.16267972250649, −10.12174156671119, −9.873847508166352, −9.413810033586018, −8.763906230129104, −8.194265856469268, −7.514880053616244, −6.793840254708567, −6.430144089947220, −5.795772921463568, −5.036746108120240, −3.905271520074007, −3.587527090102197, −2.815963163566295, −1.669006628186444, −1.171549246466970, 0,
1.171549246466970, 1.669006628186444, 2.815963163566295, 3.587527090102197, 3.905271520074007, 5.036746108120240, 5.795772921463568, 6.430144089947220, 6.793840254708567, 7.514880053616244, 8.194265856469268, 8.763906230129104, 9.413810033586018, 9.873847508166352, 10.12174156671119, 11.16267972250649, 11.63663337409932, 12.01401407528031, 12.57702017018560, 13.34071995659100, 13.96032876948664, 14.42570885794697, 14.94970375680404, 15.67507757734681, 16.29694658735132
