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
−9.976797550570743385821207153098, −9.583519632516709078886457725287, −8.506568959890215770786894958384, −7.79453831242630438227041147192, −6.66295101132503784044010015686, −5.90822318738904538272704426355, −4.91952770762883705594922981345, −3.63591260453331293383715590599, −2.14237359374816593359211641397, −0.890317407936487832552345594192,
1.84037891976303795615724343736, 3.38828178578512220070457156481, 4.09330205949848027865389545139, 5.28747811611402580297454314697, 6.53833928624663774034245696791, 7.04160403253191004132405310240, 8.189517514317924139194624854653, 9.584064037908829902041299644382, 9.866141033634750191139201468303, 10.60119065591755907700661189478