| L(s) = 1 | + (−0.964 + 2.86i)2-s + (−0.263 + 2.98i)3-s + (−4.07 − 3.09i)4-s + (5.65 + 2.25i)5-s + (−8.29 − 3.63i)6-s + (−4.32 + 2.00i)7-s + (2.80 − 1.90i)8-s + (−8.86 − 1.57i)9-s + (−11.8 + 14.0i)10-s + (−2.34 − 0.651i)11-s + (10.3 − 11.3i)12-s + (−2.95 + 5.58i)13-s + (−1.55 − 14.3i)14-s + (−8.21 + 16.2i)15-s + (−2.74 − 9.88i)16-s + (−6.05 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.482 + 1.43i)2-s + (−0.0877 + 0.996i)3-s + (−1.01 − 0.774i)4-s + (1.13 + 0.450i)5-s + (−1.38 − 0.605i)6-s + (−0.617 + 0.285i)7-s + (0.350 − 0.237i)8-s + (−0.984 − 0.174i)9-s + (−1.18 + 1.40i)10-s + (−0.213 − 0.0592i)11-s + (0.861 − 0.947i)12-s + (−0.227 + 0.429i)13-s + (−0.111 − 1.02i)14-s + (−0.547 + 1.08i)15-s + (−0.171 − 0.617i)16-s + (−0.356 + 0.770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.422223 - 0.871242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422223 - 0.871242i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.263 - 2.98i)T \) |
| 59 | \( 1 + (58.1 - 9.68i)T \) |
| good | 2 | \( 1 + (0.964 - 2.86i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-5.65 - 2.25i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (4.32 - 2.00i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (2.34 + 0.651i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (2.95 - 5.58i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (6.05 - 13.0i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-28.6 + 6.30i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (6.93 + 1.13i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-0.360 - 1.06i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-36.9 - 8.13i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (16.2 - 23.9i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (11.9 - 1.95i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-10.8 - 39.1i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-43.7 + 17.4i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (74.5 - 63.2i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (75.0 + 25.2i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-43.1 - 63.5i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-41.8 + 16.6i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-104. + 11.3i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-74.3 + 44.7i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-80.1 + 4.34i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-17.0 - 50.5i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (16.9 + 1.83i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−13.72876841648304772957832606154, −12.05012716791716187171402839404, −10.66815018062809073629557123500, −9.639082237894971016834165740349, −9.270264132519979664413123975767, −8.013500204135537913050709315459, −6.54822126302445887434990050546, −5.95812073804525562549600109494, −4.89232363813962246710791772805, −2.91138599686475021188737474265,
0.67850568843550707903059646797, 2.01284923704148123298862134196, 3.15948457130644390945570682080, 5.31637513962006707193295579023, 6.52194446626167777308577377097, 7.897777278466567266842418599838, 9.247663823540936148448638775080, 9.788299882486947045315393872501, 10.84745051325630592292238249932, 11.97439589625932749928403344778
