L(s) = 1 | + 2.89i·2-s + (−2.37 + 1.83i)3-s − 4.39·4-s + (4.55 + 7.88i)5-s + (−5.31 − 6.87i)6-s + (3.39 + 5.88i)7-s − 1.13i·8-s + (2.25 − 8.71i)9-s + (−22.8 + 13.1i)10-s + (4.36 + 7.55i)11-s + (10.4 − 8.06i)12-s − 13.8i·13-s + (−17.0 + 9.84i)14-s + (−25.2 − 10.3i)15-s − 14.2·16-s + (13.7 − 23.8i)17-s + ⋯ |
L(s) = 1 | + 1.44i·2-s + (−0.790 + 0.612i)3-s − 1.09·4-s + (0.910 + 1.57i)5-s + (−0.886 − 1.14i)6-s + (0.485 + 0.840i)7-s − 0.142i·8-s + (0.250 − 0.968i)9-s + (−2.28 + 1.31i)10-s + (0.396 + 0.687i)11-s + (0.868 − 0.672i)12-s − 1.06i·13-s + (−1.21 + 0.702i)14-s + (−1.68 − 0.689i)15-s − 0.892·16-s + (0.810 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.331568 - 1.35685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.331568 - 1.35685i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.37 - 1.83i)T \) |
| 19 | \( 1 + (-10.5 + 15.8i)T \) |
good | 2 | \( 1 - 2.89iT - 4T^{2} \) |
| 5 | \( 1 + (-4.55 - 7.88i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.39 - 5.88i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.36 - 7.55i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + (-13.7 + 23.8i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 - 21.8T + 529T^{2} \) |
| 29 | \( 1 + (17.8 + 10.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-20.2 - 11.6i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 22.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (28.9 - 16.6i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 3.79T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-23.1 + 40.1i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (34.8 - 20.1i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-34.4 + 19.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.3 - 104. i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 31.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-46.4 - 26.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (1.89 - 3.27i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + 61.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-14.5 - 25.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-10.0 + 5.78i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 126. iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.52872508943409815728309233891, −11.89278119425585839417936538135, −11.04526932886349590515605530210, −9.945524812415322908252136747067, −9.128131184797572314082685781772, −7.42354511401440575410596759845, −6.74222173904898468079719577284, −5.65806389369321415906921027323, −5.10047425758511273667138766388, −2.85299691717803693911047784644,
1.11487629342762487981184335633, 1.61966090666001447686527293691, 4.03405403736808938132669712402, 5.14957381454950167437065129932, 6.37421024430675089928077668857, 8.068337611396661738375119496482, 9.231757087839204562898532634488, 10.20387803554127747247392342841, 11.11314933223431193246593462828, 12.03809736815124117867066999668