| L(s) = 1 | + 2-s − 4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s − 2·11-s + 2·13-s − 2·14-s − 16-s − 6·17-s − 4·19-s + 2·20-s − 2·22-s + 23-s − 25-s + 2·26-s + 2·28-s + 4·31-s + 5·32-s − 6·34-s + 4·35-s + 6·37-s − 4·38-s + 6·40-s − 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s − 0.603·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 0.676·35-s + 0.986·37-s − 0.648·38-s + 0.948·40-s − 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.25038670719751, −13.07339081357685, −12.61975104442957, −12.19755069813217, −11.47215706448249, −11.29554864530521, −10.69862329020671, −10.06778895654218, −9.672106323061030, −9.027374029533381, −8.633736941463579, −8.196123500160806, −7.775213255127000, −6.913046633206295, −6.595343648096810, −6.077643772865051, −5.600266874569894, −4.809419658893146, −4.466771526184510, −4.055125808242157, −3.487065529069578, −2.952027837901324, −2.464359512981169, −1.567252550514321, −0.4825364019725469, 0,
0.4825364019725469, 1.567252550514321, 2.464359512981169, 2.952027837901324, 3.487065529069578, 4.055125808242157, 4.466771526184510, 4.809419658893146, 5.600266874569894, 6.077643772865051, 6.595343648096810, 6.913046633206295, 7.775213255127000, 8.196123500160806, 8.633736941463579, 9.027374029533381, 9.672106323061030, 10.06778895654218, 10.69862329020671, 11.29554864530521, 11.47215706448249, 12.19755069813217, 12.61975104442957, 13.07339081357685, 13.25038670719751
