| L(s) = 1 | + 2·3-s + 7-s + 9-s − 2·11-s − 13-s − 17-s − 6·19-s + 2·21-s − 9·23-s − 4·27-s − 6·29-s + 5·31-s − 4·33-s − 7·37-s − 2·39-s − 5·41-s − 8·43-s − 9·47-s + 49-s − 2·51-s + 2·53-s − 12·57-s − 14·59-s − 13·61-s + 63-s + 2·67-s − 18·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.242·17-s − 1.37·19-s + 0.436·21-s − 1.87·23-s − 0.769·27-s − 1.11·29-s + 0.898·31-s − 0.696·33-s − 1.15·37-s − 0.320·39-s − 0.780·41-s − 1.21·43-s − 1.31·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 1.58·57-s − 1.82·59-s − 1.66·61-s + 0.125·63-s + 0.244·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.59170778359681, −13.33008612384248, −12.72381413230889, −12.08064044785136, −11.91298857938117, −11.15844186990668, −10.64390910636757, −10.29013167343956, −9.728308492983214, −9.275628426863607, −8.744232428782512, −8.230966278378925, −8.004764287214387, −7.624400756817606, −6.871886310376154, −6.341557896123274, −5.903902395073888, −5.122290005145143, −4.739066987195823, −4.058786341165222, −3.613874965142329, −3.037225350792723, −2.419187961550552, −1.864484412534808, −1.614134396634640, 0, 0,
1.614134396634640, 1.864484412534808, 2.419187961550552, 3.037225350792723, 3.613874965142329, 4.058786341165222, 4.739066987195823, 5.122290005145143, 5.903902395073888, 6.341557896123274, 6.871886310376154, 7.624400756817606, 8.004764287214387, 8.230966278378925, 8.744232428782512, 9.275628426863607, 9.728308492983214, 10.29013167343956, 10.64390910636757, 11.15844186990668, 11.91298857938117, 12.08064044785136, 12.72381413230889, 13.33008612384248, 13.59170778359681
