L(s) = 1 | + (−1 + 1.73i)2-s + (−0.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (1 − 1.73i)5-s + 1.99·6-s + (−2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (1.99 + 3.46i)10-s + (1 + 1.73i)11-s + (−1 + 1.73i)12-s + 13-s + (1.00 − 5.19i)14-s − 1.99·15-s + (1.99 − 3.46i)16-s + (−0.999 − 1.73i)18-s + (−0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.447 − 0.774i)5-s + 0.816·6-s + (−0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.632 + 1.09i)10-s + (0.301 + 0.522i)11-s + (−0.288 + 0.499i)12-s + 0.277·13-s + (0.267 − 1.38i)14-s − 0.516·15-s + (0.499 − 0.866i)16-s + (−0.235 − 0.408i)18-s + (−0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.389127 + 0.192901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.389127 + 0.192901i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 2 | \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−18.12129056260894093908436548569, −17.04031378858136437073408927109, −16.32959379130278819697921305148, −15.07414260816926352661587808986, −13.34323482627396490642499025152, −12.15250100734095017521574237444, −9.745945150935527986080392896027, −8.601280514767607011404874617841, −6.94429713931444545466421079467, −5.69031773444673653876183633556,
3.25829500936861479080741427724, 6.38832905030081022881130156037, 9.036766158069173391425947412212, 10.22619556082021660311570594860, 10.97701824034660861287431573674, 12.46535281282447583047007665638, 14.09592991053196328885893654920, 15.79761123923961004510355109668, 17.20622136474437902816441882031, 18.35162599690320300427413645604