| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 4·7-s + 8-s + 6·9-s − 2·11-s − 3·12-s − 13-s − 4·14-s + 16-s + 6·18-s − 7·19-s + 12·21-s − 2·22-s − 6·23-s − 3·24-s − 26-s − 9·27-s − 4·28-s + 3·29-s − 7·31-s + 32-s + 6·33-s + 6·36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s − 0.603·11-s − 0.866·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.41·18-s − 1.60·19-s + 2.61·21-s − 0.426·22-s − 1.25·23-s − 0.612·24-s − 0.196·26-s − 1.73·27-s − 0.755·28-s + 0.557·29-s − 1.25·31-s + 0.176·32-s + 1.04·33-s + 36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.59044229023270, −16.10225375937617, −15.78064374636608, −15.15399612324722, −14.50247067569712, −13.58559370314168, −13.07295568573968, −12.68010567816344, −12.28635634142310, −11.73750784493917, −11.01275532650162, −10.58043868239619, −9.977234918454434, −9.666660208186801, −8.520670574744076, −7.769942500055195, −6.828946914941195, −6.464079387157040, −6.238389106012990, −5.248355581690459, −5.051913655466616, −4.035436758353447, −3.559481590924168, −2.503417637346489, −1.613534160680452, 0, 0,
1.613534160680452, 2.503417637346489, 3.559481590924168, 4.035436758353447, 5.051913655466616, 5.248355581690459, 6.238389106012990, 6.464079387157040, 6.828946914941195, 7.769942500055195, 8.520670574744076, 9.666660208186801, 9.977234918454434, 10.58043868239619, 11.01275532650162, 11.73750784493917, 12.28635634142310, 12.68010567816344, 13.07295568573968, 13.58559370314168, 14.50247067569712, 15.15399612324722, 15.78064374636608, 16.10225375937617, 16.59044229023270
