| L(s) = 1 | + (2.97 − 2.16i)2-s + (4.28 − 2.94i)3-s + (1.70 − 5.24i)4-s + (−10.1 + 13.9i)5-s + (6.38 − 18.0i)6-s + (−16.0 − 5.21i)7-s + (2.82 + 8.70i)8-s + (9.69 − 25.1i)9-s + 63.2i·10-s + (36.4 − 0.331i)11-s + (−8.11 − 27.4i)12-s + (−34.3 − 47.2i)13-s + (−59.0 + 19.1i)14-s + (−2.38 + 89.4i)15-s + (62.8 + 45.6i)16-s + (−1.44 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (1.05 − 0.763i)2-s + (0.824 − 0.566i)3-s + (0.212 − 0.655i)4-s + (−0.905 + 1.24i)5-s + (0.434 − 1.22i)6-s + (−0.867 − 0.281i)7-s + (0.124 + 0.384i)8-s + (0.359 − 0.933i)9-s + 2.00i·10-s + (0.999 − 0.00908i)11-s + (−0.195 − 0.660i)12-s + (−0.732 − 1.00i)13-s + (−1.12 + 0.366i)14-s + (−0.0410 + 1.53i)15-s + (0.982 + 0.713i)16-s + (−0.0206 − 0.0149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.89599 - 0.904804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89599 - 0.904804i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.28 + 2.94i)T \) |
| 11 | \( 1 + (-36.4 + 0.331i)T \) |
| good | 2 | \( 1 + (-2.97 + 2.16i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (10.1 - 13.9i)T + (-38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (16.0 + 5.21i)T + (277. + 201. i)T^{2} \) |
| 13 | \( 1 + (34.3 + 47.2i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (1.44 + 1.05i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (22.6 - 7.35i)T + (5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 75.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-17.7 + 54.5i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-127. + 92.9i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (71.2 - 219. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (50.3 + 154. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 128. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (404. - 131. i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-357. - 492. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (13.0 + 4.23i)T + (1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-455. + 626. i)T + (-7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 199.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (304. - 418. i)T + (-1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-930. - 302. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (486. + 669. i)T + (-1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-443. - 322. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 399. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (717. - 521. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.35955925284234418091993071736, −14.61522932598361066920579183424, −13.59011122868679582422998216743, −12.47353558695288736790390084791, −11.50571316827350810384735983053, −10.00619966012322696544410093775, −7.919045274774202063652683982059, −6.62021025151045956301235593227, −3.80520027502689347425150235771, −2.88869007775586541087724832811,
3.84892063279643382285790673381, 4.83902511651706330803983162769, 6.86919074240482039174424965310, 8.571565176201064749376012668417, 9.658669707271825406117471326026, 12.04654349153605282356401351730, 12.97943748555179547140958435321, 14.22705353975509790874966914224, 15.16056162069925470181916248203, 16.25895713414696321338377447398
