| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 5.19i·5-s − 8.34·7-s − 2.82i·8-s − 7.34·10-s + 0.953i·11-s − 9.69·13-s − 11.8i·14-s + 4.00·16-s + 18.8i·17-s − 24.6·19-s − 10.3i·20-s − 1.34·22-s − 0.953i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.03i·5-s − 1.19·7-s − 0.353i·8-s − 0.734·10-s + 0.0866i·11-s − 0.745·13-s − 0.843i·14-s + 0.250·16-s + 1.11i·17-s − 1.29·19-s − 0.519i·20-s − 0.0612·22-s − 0.0414i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(-0.729320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.729320i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 7 | \( 1 + 8.34T + 49T^{2} \) |
| 11 | \( 1 - 0.953iT - 121T^{2} \) |
| 13 | \( 1 + 9.69T + 169T^{2} \) |
| 17 | \( 1 - 18.8iT - 289T^{2} \) |
| 19 | \( 1 + 24.6T + 361T^{2} \) |
| 23 | \( 1 + 0.953iT - 529T^{2} \) |
| 29 | \( 1 - 13.6iT - 841T^{2} \) |
| 31 | \( 1 - 3.04T + 961T^{2} \) |
| 37 | \( 1 - 46.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 45.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 94.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 75.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.90T + 5.32e3T^{2} \) |
| 79 | \( 1 + 43.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 130. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−13.08917380582722791529734655618, −12.46620534075010599367933865948, −10.86736018320948566105403356828, −10.09019040552175666574333891322, −9.047720710102833870710738179347, −7.70588492282932355660362256315, −6.66525936196583481203423502646, −6.02265701632888676318450386899, −4.21335890700690418559036555164, −2.80290026511601776260126141880,
0.43740435151429851816795116332, 2.55596884653597274144918604449, 4.11882920190032061259256842791, 5.30326985027717009797753548261, 6.74848135309114683620825560554, 8.266578175984212976841839558238, 9.333850070369221699523705877105, 9.911822741761735684439051415260, 11.23396147625797902056331519499, 12.37146113146669549156111230568
