| L(s) = 1 | + 2-s − 4-s + 5-s + 2·7-s − 3·8-s + 10-s − 4·11-s + 13-s + 2·14-s − 16-s − 8·19-s − 20-s − 4·22-s − 8·23-s + 25-s + 26-s − 2·28-s − 6·29-s + 4·31-s + 5·32-s + 2·35-s − 8·37-s − 8·38-s − 3·40-s − 10·41-s + 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.277·13-s + 0.534·14-s − 1/4·16-s − 1.83·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.338·35-s − 1.31·37-s − 1.29·38-s − 0.474·40-s − 1.56·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.76412779267614, −13.29982239618508, −12.79234845542069, −12.39476035754940, −12.01673581232869, −11.38595350641822, −10.75722128597189, −10.41897541028961, −10.08477909016587, −9.287800938882044, −8.911503955103048, −8.415421844854563, −7.872864387584783, −7.668772639640817, −6.644521887844549, −6.147268593980035, −5.917892220956907, −5.104981257899925, −4.895851348939740, −4.369226267583554, −3.716524454266480, −3.294696117502094, −2.351458579394375, −2.094366144955035, −1.351484042560131, 0, 0,
1.351484042560131, 2.094366144955035, 2.351458579394375, 3.294696117502094, 3.716524454266480, 4.369226267583554, 4.895851348939740, 5.104981257899925, 5.917892220956907, 6.147268593980035, 6.644521887844549, 7.668772639640817, 7.872864387584783, 8.415421844854563, 8.911503955103048, 9.287800938882044, 10.08477909016587, 10.41897541028961, 10.75722128597189, 11.38595350641822, 12.01673581232869, 12.39476035754940, 12.79234845542069, 13.29982239618508, 13.76412779267614
