| L(s) = 1 | + (−0.460 + 1.71i)2-s + (1.57 + 2.72i)3-s + (0.726 + 0.419i)4-s + (−5.39 + 1.44i)6-s + (1.30 + 4.87i)7-s + (−6.08 + 6.08i)8-s + (−0.437 + 0.758i)9-s + (−3.62 − 0.970i)11-s + 2.63i·12-s + (−12.9 − 0.340i)13-s − 8.97·14-s + (−5.97 − 10.3i)16-s + (23.0 + 13.2i)17-s + (−1.10 − 1.10i)18-s + (−15.0 + 4.03i)19-s + ⋯ |
| L(s) = 1 | + (−0.230 + 0.858i)2-s + (0.523 + 0.907i)3-s + (0.181 + 0.104i)4-s + (−0.899 + 0.241i)6-s + (0.186 + 0.696i)7-s + (−0.760 + 0.760i)8-s + (−0.0486 + 0.0842i)9-s + (−0.329 − 0.0882i)11-s + 0.219i·12-s + (−0.999 − 0.0261i)13-s − 0.641·14-s + (−0.373 − 0.646i)16-s + (1.35 + 0.781i)17-s + (−0.0611 − 0.0611i)18-s + (−0.792 + 0.212i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0105556 - 1.69487i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0105556 - 1.69487i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (12.9 + 0.340i)T \) |
| good | 2 | \( 1 + (0.460 - 1.71i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (-1.57 - 2.72i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 4.87i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (3.62 + 0.970i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-23.0 - 13.2i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 - 4.03i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (6.64 - 3.83i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-7.79 - 13.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-7.36 - 7.36i)T + 961iT^{2} \) |
| 37 | \( 1 + (-0.251 - 0.0673i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-5.39 + 20.1i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (6.34 + 3.66i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-57.7 + 57.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 25.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (14.8 + 55.5i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (27.1 - 47.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (19.7 - 73.6i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-37.8 + 10.1i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (71.8 - 71.8i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 123.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-82.4 - 82.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-93.8 - 25.1i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (126. - 34.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−12.03859208496077476705546319453, −10.64699675397015427104575615098, −9.854384727220149220675784819568, −8.825926526167415531544816870074, −8.197927067062904967372052564676, −7.19341471162065158250088294393, −5.97499340884601432392643481573, −5.06086011319069604268284037088, −3.60917121623007596105378073096, −2.41320415653265102658284668100,
0.793753245963854660074581440998, 2.09745431802657856583066223514, 3.03468605588629856574428243981, 4.64263919627945751936135968032, 6.22229766459243123982718153640, 7.35296648566133125188982710654, 7.84069791069668727543676407711, 9.260837788917166112367384042137, 10.16041082886754239284338662482, 10.82948828415341576460481465614
