| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 15-s + 5·17-s + 21-s − 23-s − 4·25-s − 27-s − 2·31-s − 33-s + 35-s + 10·37-s + 2·41-s − 43-s − 45-s + 9·47-s − 6·49-s − 5·51-s − 2·53-s − 55-s − 6·59-s + 5·61-s − 63-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.258·15-s + 1.21·17-s + 0.218·21-s − 0.208·23-s − 4/5·25-s − 0.192·27-s − 0.359·31-s − 0.174·33-s + 0.169·35-s + 1.64·37-s + 0.312·41-s − 0.152·43-s − 0.149·45-s + 1.31·47-s − 6/7·49-s − 0.700·51-s − 0.274·53-s − 0.134·55-s − 0.781·59-s + 0.640·61-s − 0.125·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 398544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 398544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−12.59803223616417, −12.16550545127222, −11.74440166776164, −11.40391191526426, −10.93151894375248, −10.43694065989744, −9.914272678600358, −9.603039876940640, −9.183156491922794, −8.545851431641626, −7.927658282897310, −7.705682834462557, −7.210449978925811, −6.639306974926802, −6.125595705459412, −5.786107277180517, −5.316530497198196, −4.684193422286638, −4.163792669232739, −3.755479650266567, −3.233099180927805, −2.627732886788091, −1.956238137870871, −1.240078494167161, −0.7192159650689911, 0,
0.7192159650689911, 1.240078494167161, 1.956238137870871, 2.627732886788091, 3.233099180927805, 3.755479650266567, 4.163792669232739, 4.684193422286638, 5.316530497198196, 5.786107277180517, 6.125595705459412, 6.639306974926802, 7.210449978925811, 7.705682834462557, 7.927658282897310, 8.545851431641626, 9.183156491922794, 9.603039876940640, 9.914272678600358, 10.43694065989744, 10.93151894375248, 11.40391191526426, 11.74440166776164, 12.16550545127222, 12.59803223616417
