| L(s) = 1 | + (−0.182 + 0.528i)2-s + (0.506 + 0.982i)3-s + (1.32 + 1.04i)4-s + (2.49 − 0.237i)5-s + (−0.611 + 0.0879i)6-s + (−2.11 − 1.58i)7-s + (−1.73 + 1.11i)8-s + (1.03 − 1.44i)9-s + (−0.329 + 1.35i)10-s + (−3.07 + 1.06i)11-s + (−0.352 + 1.83i)12-s + (−0.700 − 2.38i)13-s + (1.22 − 0.826i)14-s + (1.49 + 2.32i)15-s + (0.523 + 2.15i)16-s + (3.83 + 1.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.129 + 0.373i)2-s + (0.292 + 0.567i)3-s + (0.663 + 0.521i)4-s + (1.11 − 0.106i)5-s + (−0.249 + 0.0359i)6-s + (−0.799 − 0.600i)7-s + (−0.613 + 0.394i)8-s + (0.343 − 0.482i)9-s + (−0.104 + 0.430i)10-s + (−0.927 + 0.321i)11-s + (−0.101 + 0.528i)12-s + (−0.194 − 0.661i)13-s + (0.327 − 0.220i)14-s + (0.386 + 0.600i)15-s + (0.130 + 0.539i)16-s + (0.929 + 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20484 + 0.683053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20484 + 0.683053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.11 + 1.58i)T \) |
| 23 | \( 1 + (0.461 + 4.77i)T \) |
| good | 2 | \( 1 + (0.182 - 0.528i)T + (-1.57 - 1.23i)T^{2} \) |
| 3 | \( 1 + (-0.506 - 0.982i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (-2.49 + 0.237i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (3.07 - 1.06i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.700 + 2.38i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.83 - 1.53i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (6.46 - 2.58i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.213 + 1.48i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.07 + 0.0988i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (9.82 + 6.99i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-7.12 - 3.25i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.14 + 3.34i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.13 + 0.655i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.71 - 5.99i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-7.10 - 1.72i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (2.13 + 1.09i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (2.43 + 12.6i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-2.81 - 3.24i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (3.69 - 4.69i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (7.65 - 8.02i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.91 - 4.19i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.504 - 10.5i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (0.409 - 0.895i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−12.73530891565685202586477348652, −12.55063625431527468820077041927, −10.40161981136816699304446904471, −10.29074743692821121736508447831, −8.999357881523383351885828262499, −7.81514125708767934233368113857, −6.63797919214502025846909218163, −5.70722120450127252277539425468, −3.88558515855932741213944147585, −2.51202424341020329598635930307,
1.91448356529852661433669398003, 2.80732663085514829031521271637, 5.35591526319455034731465173866, 6.30544661732388449734651911753, 7.26894712983473518291070902164, 8.811914388833006178883227261819, 9.917485476329511827305765589057, 10.46130677197982395288790274662, 11.77034703543435125854103962520, 12.82943970696226207295433927138
