| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 4·7-s + 4·10-s + 11-s + 4·13-s − 8·14-s − 4·16-s + 4·17-s − 6·19-s − 4·20-s − 2·22-s − 23-s − 25-s − 8·26-s + 8·28-s + 31-s + 8·32-s − 8·34-s − 8·35-s + 3·37-s + 12·38-s − 2·41-s + 12·43-s + 2·44-s + 2·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 1.51·7-s + 1.26·10-s + 0.301·11-s + 1.10·13-s − 2.13·14-s − 16-s + 0.970·17-s − 1.37·19-s − 0.894·20-s − 0.426·22-s − 0.208·23-s − 1/5·25-s − 1.56·26-s + 1.51·28-s + 0.179·31-s + 1.41·32-s − 1.37·34-s − 1.35·35-s + 0.493·37-s + 1.94·38-s − 0.312·41-s + 1.82·43-s + 0.301·44-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7912355108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7912355108\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−10.14699813320344871568896778184, −9.042720857679721134052233248892, −8.281922090633992161077459278719, −8.020557649607552611162113556218, −7.16792216484054517737503154680, −5.98757410762844504568083159424, −4.64044667538893438397726921426, −3.83590502335216880130516763272, −2.02279147792362879497590466766, −0.931362933455762091067918970399,
0.931362933455762091067918970399, 2.02279147792362879497590466766, 3.83590502335216880130516763272, 4.64044667538893438397726921426, 5.98757410762844504568083159424, 7.16792216484054517737503154680, 8.020557649607552611162113556218, 8.281922090633992161077459278719, 9.042720857679721134052233248892, 10.14699813320344871568896778184
