L(s) = 1 | + 1.73i·2-s − 1.73i·3-s − 0.999·4-s + (1.5 + 2.59i)5-s + 2.99·6-s + 1.73i·8-s − 2.99·9-s + (−4.5 + 2.59i)10-s + (1.5 + 0.866i)11-s + 1.73i·12-s + (−1.5 − 0.866i)13-s + (4.5 − 2.59i)15-s − 5·16-s + (1.5 + 2.59i)17-s − 5.19i·18-s + (4.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + 1.22i·2-s − 0.999i·3-s − 0.499·4-s + (0.670 + 1.16i)5-s + 1.22·6-s + 0.612i·8-s − 0.999·9-s + (−1.42 + 0.821i)10-s + (0.452 + 0.261i)11-s + 0.499i·12-s + (−0.416 − 0.240i)13-s + (1.16 − 0.670i)15-s − 1.25·16-s + (0.363 + 0.630i)17-s − 1.22i·18-s + (1.03 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975372 + 1.24017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975372 + 1.24017i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 2.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-7.5 + 4.33i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.40632432733225108182000205329, −10.50328884458875860361275540767, −9.367751134025422031806863187691, −8.240041641280609738669055701456, −7.37493504212361249897340938760, −6.80947594156380770543209143256, −6.06458625881309539539596987073, −5.25768247432087086051004324024, −3.17040035285208912741599036083, −1.94081127087227006539882941294,
1.08390794351730416373759210551, 2.64444307993536269557638908476, 3.78191481176377482405687732830, 4.87761163420525156942783335090, 5.63165539068504427278782897037, 7.20026455941060238563394181494, 8.841072049975956968599589901107, 9.347695112760174966469296447698, 9.861328490816099585712786417665, 10.92516608470411466306902315438