| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s − 3·9-s + 10-s + 11-s − 7·13-s − 4·14-s + 16-s − 3·17-s − 3·18-s − 2·19-s + 20-s + 22-s − 23-s + 25-s − 7·26-s − 4·28-s − 9·29-s + 2·31-s + 32-s − 3·34-s − 4·35-s − 3·36-s − 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s − 9-s + 0.316·10-s + 0.301·11-s − 1.94·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 0.208·23-s + 1/5·25-s − 1.37·26-s − 0.755·28-s − 1.67·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s − 0.676·35-s − 1/2·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 419 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.87053838484773, −14.76742062190708, −14.03537812816098, −13.47391004602085, −13.24856954022869, −12.45453353611310, −12.20062097314215, −11.79917161659655, −10.86513892529237, −10.62203910159092, −9.793596016450536, −9.477890861743968, −9.016316975812893, −8.298309335936061, −7.455229816469062, −7.012097200932346, −6.484091856802157, −5.987345733996505, −5.420926826104092, −4.879599201344138, −4.148937030355437, −3.427890930368043, −2.885279312054159, −2.373187705157199, −1.699215812977932, 0, 0,
1.699215812977932, 2.373187705157199, 2.885279312054159, 3.427890930368043, 4.148937030355437, 4.879599201344138, 5.420926826104092, 5.987345733996505, 6.484091856802157, 7.012097200932346, 7.455229816469062, 8.298309335936061, 9.016316975812893, 9.477890861743968, 9.793596016450536, 10.62203910159092, 10.86513892529237, 11.79917161659655, 12.20062097314215, 12.45453353611310, 13.24856954022869, 13.47391004602085, 14.03537812816098, 14.76742062190708, 14.87053838484773
