L(s) = 1 | + (0.583 + 0.336i)2-s + (−1.77 − 3.07i)4-s + (−2.27 + 3.93i)5-s + 9.87·7-s − 5.08i·8-s + (−2.65 + 1.53i)10-s + 15.6·11-s + (13.3 − 7.68i)13-s + (5.76 + 3.32i)14-s + (−5.37 + 9.31i)16-s + (12.4 − 21.5i)17-s + (−18.4 + 4.63i)19-s + 16.1·20-s + (9.12 + 5.26i)22-s + (−4.15 − 7.19i)23-s + ⋯ |
L(s) = 1 | + (0.291 + 0.168i)2-s + (−0.443 − 0.767i)4-s + (−0.454 + 0.787i)5-s + 1.41·7-s − 0.635i·8-s + (−0.265 + 0.153i)10-s + 1.42·11-s + (1.02 − 0.590i)13-s + (0.411 + 0.237i)14-s + (−0.336 + 0.582i)16-s + (0.730 − 1.26i)17-s + (−0.969 + 0.243i)19-s + 0.806·20-s + (0.414 + 0.239i)22-s + (−0.180 − 0.312i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74684 - 0.157007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74684 - 0.157007i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + (18.4 - 4.63i)T \) |
good | 2 | \( 1 + (-0.583 - 0.336i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 + (2.27 - 3.93i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 - 9.87T + 49T^{2} \) |
| 11 | \( 1 - 15.6T + 121T^{2} \) |
| 13 | \( 1 + (-13.3 + 7.68i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-12.4 + 21.5i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (4.15 + 7.19i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (27.3 - 15.7i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 30.9iT - 961T^{2} \) |
| 37 | \( 1 + 17.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-44.2 - 25.5i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (0.773 - 1.33i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (5.09 + 8.81i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (4.54 - 2.62i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (68.4 + 39.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (53.9 + 93.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (16.9 - 9.80i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-45.3 - 26.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (36.6 - 63.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (27.1 + 15.6i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (114. - 66.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (70.2 + 40.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.47525206836194982495967469868, −11.21926542345912141250593644024, −10.84168167610591961841552373104, −9.445107718834016117420445779315, −8.397288083990154984317569720220, −7.15967867952649282589927613363, −6.02921277364822162061382557859, −4.80904916845132666853380906035, −3.64079735509608469227193632118, −1.33810331491247913736206653133,
1.57157107967013314014852328612, 3.99083230508453840355131077720, 4.37825163417346417460519279224, 5.97112151800625340654194573376, 7.71155833066377134056235916287, 8.483044762553762325971569005543, 9.116685212894932215487994443241, 10.98086748977877295860602396903, 11.72636815232303596507437832560, 12.41348140605298236936369565036