| 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
−11.04528626467816196222628216520, −9.786703950810431867018218624751, −9.186921235050006874979357764409, −8.302809833772119357370343462956, −7.38073981783995063236889047649, −6.47311095032860563333794808235, −5.25852859863971258178967154226, −3.85488937086871057402423251383, −2.78060042448644732186697579411, −1.23636189934717871891265306193,
2.02459197350193385452114081769, 3.31156072543172373401061063630, 4.18555397680291478596828099231, 5.57925803135726359963553338490, 6.70197445915838435107256322637, 7.76528774574969640105612500001, 8.658822662809098178165060861050, 9.504288384990632913303892722727, 10.36748912497910014941959377900, 10.95462774239024642211088390938
