L(s) = 1 | − 0.414i·2-s + 1.82·4-s + 2.93·5-s − 1.58i·8-s − 1.21i·10-s − 4.82i·11-s + 2.93i·13-s + 3·16-s − 7.07·17-s + 5.86i·19-s + 5.35·20-s − 1.99·22-s + 2i·23-s + 3.58·25-s + 1.21·26-s + ⋯ |
L(s) = 1 | − 0.292i·2-s + 0.914·4-s + 1.31·5-s − 0.560i·8-s − 0.383i·10-s − 1.45i·11-s + 0.812i·13-s + 0.750·16-s − 1.71·17-s + 1.34i·19-s + 1.19·20-s − 0.426·22-s + 0.417i·23-s + 0.717·25-s + 0.238·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95595 - 0.534847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95595 - 0.534847i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 5.86iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 0.828iT - 29T^{2} \) |
| 31 | \( 1 + 5.86iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 5.86T + 59T^{2} \) |
| 61 | \( 1 - 1.21iT - 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 0.828iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−11.03123987990486918242385734858, −10.29033016986047027858086220981, −9.351230968105701448367265067997, −8.493197555373189547438942973327, −7.14447339648659723823607067696, −6.16664078910544433489574451100, −5.72023626143805448293462675793, −3.96453924116737751355353249677, −2.59586479729940079319977276241, −1.61764564249852980721179987264,
1.90607441545801680421054824736, 2.67463167110657769592876196361, 4.66692034548636840498003141407, 5.60644107097189058033372792074, 6.71484269538740013911907216201, 7.10987753088453204996606162631, 8.523349353286868070949716133981, 9.460037519816967523724871735098, 10.38046968805263656967104027830, 10.96798398471856688161771092837