L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 8.60i·5-s − 2.82i·8-s + 12.1·10-s − 2.82i·11-s − 12.1·13-s + 4.00·16-s − 25.8i·17-s + 24.3·19-s + 17.2i·20-s + 4.00·22-s + 42.4i·23-s − 49·25-s − 17.2i·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.72i·5-s − 0.353i·8-s + 1.21·10-s − 0.257i·11-s − 0.935·13-s + 0.250·16-s − 1.51i·17-s + 1.28·19-s + 0.860i·20-s + 0.181·22-s + 1.84i·23-s − 1.95·25-s − 0.661i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6476952475\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6476952475\) |
\(L(2)\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 8.60iT - 25T^{2} \) |
| 11 | \( 1 + 2.82iT - 121T^{2} \) |
| 13 | \( 1 + 12.1T + 169T^{2} \) |
| 17 | \( 1 + 25.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.3T + 361T^{2} \) |
| 23 | \( 1 - 42.4iT - 529T^{2} \) |
| 29 | \( 1 + 15.5iT - 841T^{2} \) |
| 31 | \( 1 + 24.3T + 961T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 97.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 104T + 4.48e3T^{2} \) |
| 71 | \( 1 + 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 94.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 158.T + 9.40e3T^{2} \) |
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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
−9.305147565829497401559365771471, −8.939651097246258510497194214267, −7.64171679816264929049934802563, −7.43589090533826086707910553609, −5.85164225374559524640314830051, −5.15951976411244748106373266657, −4.62880457375277762205533627978, −3.27215240928678424396207865802, −1.44949666130378794019637614118, −0.21251314587217680263919261291,
1.85510960251154594786740767522, 2.85537934973616689009891257080, 3.61632056576438744305955260752, 4.80901252622570268956694178078, 6.04057963136690073635759412224, 6.89810729838789633614437076043, 7.65586115506888548046826406500, 8.688499754477104325322466232888, 9.864841267880452423455848858897, 10.33725631629655156777986851961