| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·5-s − 2·6-s − 2·7-s + 9-s + 4·10-s − 11-s + 2·12-s + 7·13-s + 4·14-s − 2·15-s − 4·16-s − 4·17-s − 2·18-s − 4·20-s − 2·21-s + 2·22-s − 6·23-s − 25-s − 14·26-s + 27-s − 4·28-s − 29-s + 4·30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.577·12-s + 1.94·13-s + 1.06·14-s − 0.516·15-s − 16-s − 0.970·17-s − 0.471·18-s − 0.894·20-s − 0.436·21-s + 0.426·22-s − 1.25·23-s − 1/5·25-s − 2.74·26-s + 0.192·27-s − 0.755·28-s − 0.185·29-s + 0.730·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 191 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−10.02178740155305455309327116177, −9.341614754610731188862757881261, −8.389386763353821536588534213733, −8.089515968377682611906770669830, −7.02284768449059301743660948217, −6.12160767971106805790952621807, −4.25421800965857265960724529460, −3.37428893344660676971528492086, −1.76107484759245971052768904108, 0,
1.76107484759245971052768904108, 3.37428893344660676971528492086, 4.25421800965857265960724529460, 6.12160767971106805790952621807, 7.02284768449059301743660948217, 8.089515968377682611906770669830, 8.389386763353821536588534213733, 9.341614754610731188862757881261, 10.02178740155305455309327116177
