L(s) = 1 | + (1.48 + 0.898i)3-s + (0.357 + 0.982i)5-s + (−0.862 + 0.152i)7-s + (1.38 + 2.66i)9-s + (0.855 + 0.311i)11-s + (0.662 + 0.555i)13-s + (−0.353 + 1.77i)15-s + (4.52 + 2.61i)17-s + (−3.63 + 2.09i)19-s + (−1.41 − 0.550i)21-s + (0.356 − 2.02i)23-s + (2.99 − 2.51i)25-s + (−0.341 + 5.18i)27-s + (0.526 + 0.627i)29-s + (−8.52 − 1.50i)31-s + ⋯ |
L(s) = 1 | + (0.854 + 0.518i)3-s + (0.159 + 0.439i)5-s + (−0.326 + 0.0575i)7-s + (0.461 + 0.887i)9-s + (0.257 + 0.0938i)11-s + (0.183 + 0.154i)13-s + (−0.0912 + 0.458i)15-s + (1.09 + 0.633i)17-s + (−0.834 + 0.481i)19-s + (−0.308 − 0.120i)21-s + (0.0743 − 0.421i)23-s + (0.598 − 0.502i)25-s + (−0.0657 + 0.997i)27-s + (0.0977 + 0.116i)29-s + (−1.53 − 0.269i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60148 + 0.916446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60148 + 0.916446i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.48 - 0.898i)T \) |
good | 5 | \( 1 + (-0.357 - 0.982i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.862 - 0.152i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.855 - 0.311i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.662 - 0.555i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.52 - 2.61i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.63 - 2.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.356 + 2.02i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.526 - 0.627i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.52 + 1.50i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.04 + 6.00i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.0722 + 0.198i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.30 + 7.38i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 + (-11.8 + 4.29i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.82 + 10.3i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.37 - 1.63i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.76 + 3.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.656 + 1.13i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.814 + 0.970i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.32 + 7.82i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.27 + 2.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.6 + 4.95i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.95462774239024642211088390938, −10.36748912497910014941959377900, −9.504288384990632913303892722727, −8.658822662809098178165060861050, −7.76528774574969640105612500001, −6.70197445915838435107256322637, −5.57925803135726359963553338490, −4.18555397680291478596828099231, −3.31156072543172373401061063630, −2.02459197350193385452114081769,
1.23636189934717871891265306193, 2.78060042448644732186697579411, 3.85488937086871057402423251383, 5.25852859863971258178967154226, 6.47311095032860563333794808235, 7.38073981783995063236889047649, 8.302809833772119357370343462956, 9.186921235050006874979357764409, 9.786703950810431867018218624751, 11.04528626467816196222628216520