L(s) = 1 | + 1.28i·3-s + (0.841 + 2.07i)5-s + (−1.13 − 1.13i)7-s + 1.35·9-s + (2.32 + 2.32i)11-s − 1.36·13-s + (−2.65 + 1.07i)15-s + (5.25 + 5.25i)17-s + (−3.69 − 3.69i)19-s + (1.46 − 1.46i)21-s + (−0.911 + 0.911i)23-s + (−3.58 + 3.48i)25-s + 5.58i·27-s + (−2.37 + 2.37i)29-s − 0.242i·31-s + ⋯ |
L(s) = 1 | + 0.739i·3-s + (0.376 + 0.926i)5-s + (−0.430 − 0.430i)7-s + 0.452·9-s + (0.700 + 0.700i)11-s − 0.378·13-s + (−0.685 + 0.278i)15-s + (1.27 + 1.27i)17-s + (−0.848 − 0.848i)19-s + (0.318 − 0.318i)21-s + (−0.189 + 0.189i)23-s + (−0.716 + 0.697i)25-s + 1.07i·27-s + (−0.440 + 0.440i)29-s − 0.0435i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.990305 + 1.17803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.990305 + 1.17803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.841 - 2.07i)T \) |
good | 3 | \( 1 - 1.28iT - 3T^{2} \) |
| 7 | \( 1 + (1.13 + 1.13i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.32 - 2.32i)T + 11iT^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + (-5.25 - 5.25i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.69 + 3.69i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.911 - 0.911i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.37 - 2.37i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.242iT - 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 2.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.04T + 43T^{2} \) |
| 47 | \( 1 + (7.87 - 7.87i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.80iT - 53T^{2} \) |
| 59 | \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.67 - 6.67i)T + 61iT^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (-1.49 - 1.49i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.26iT - 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + (1.63 + 1.63i)T + 97iT^{2} \) |
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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
−10.60119065591755907700661189478, −9.866141033634750191139201468303, −9.584064037908829902041299644382, −8.189517514317924139194624854653, −7.04160403253191004132405310240, −6.53833928624663774034245696791, −5.28747811611402580297454314697, −4.09330205949848027865389545139, −3.38828178578512220070457156481, −1.84037891976303795615724343736,
0.890317407936487832552345594192, 2.14237359374816593359211641397, 3.63591260453331293383715590599, 4.91952770762883705594922981345, 5.90822318738904538272704426355, 6.66295101132503784044010015686, 7.79453831242630438227041147192, 8.506568959890215770786894958384, 9.583519632516709078886457725287, 9.976797550570743385821207153098