| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·10-s + 11-s + 13-s − 4·16-s − 17-s + 2·19-s + 2·20-s − 2·22-s + 3·23-s + 25-s − 2·26-s + 2·29-s + 6·31-s + 8·32-s + 2·34-s + 11·37-s − 4·38-s − 5·41-s + 4·43-s + 2·44-s − 6·46-s − 10·47-s − 2·50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s + 0.301·11-s + 0.277·13-s − 16-s − 0.242·17-s + 0.458·19-s + 0.447·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s + 1.07·31-s + 1.41·32-s + 0.342·34-s + 1.80·37-s − 0.648·38-s − 0.780·41-s + 0.609·43-s + 0.301·44-s − 0.884·46-s − 1.45·47-s − 0.282·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−15.75560632691115, −15.05162633374122, −14.39523574355589, −13.90578026979464, −13.29808716286294, −12.87887548729049, −12.10685625836223, −11.33976425325969, −11.18557019699735, −10.45852968394772, −9.809784303977874, −9.593856100144629, −9.005240309123335, −8.335101910039835, −8.030976675674588, −7.304046337183989, −6.632579220646870, −6.309596053638722, −5.415127606644635, −4.694075782972297, −4.114344915845229, −3.024220630417244, −2.510408555300152, −1.474211353926216, −1.100054703307495, 0,
1.100054703307495, 1.474211353926216, 2.510408555300152, 3.024220630417244, 4.114344915845229, 4.694075782972297, 5.415127606644635, 6.309596053638722, 6.632579220646870, 7.304046337183989, 8.030976675674588, 8.335101910039835, 9.005240309123335, 9.593856100144629, 9.809784303977874, 10.45852968394772, 11.18557019699735, 11.33976425325969, 12.10685625836223, 12.87887548729049, 13.29808716286294, 13.90578026979464, 14.39523574355589, 15.05162633374122, 15.75560632691115
