| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16282 - 0.00445881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16282 - 0.00445881i\) |
| \(L(\frac{3}{2})\) |
|
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
−11.07892120339953751005632500952, −10.52206846979148581323376202366, −9.528141802817980089630858384989, −8.138145934635975834788744060516, −7.41673048897383049617155709409, −6.38468037509057893514556038171, −5.58005559921625378235713997472, −4.49980076520728393385753502776, −2.87054677160541241992766279726, −1.24391346617201218073583105343,
1.08782610840144141621294478234, 3.17719122742178025301244798724, 4.58701077653980968313653486547, 5.30898003607939890176247631589, 6.22428663993248955358208795390, 7.44582026513677160303273777122, 8.470951374343649779727692119833, 9.657708245376956399685030548412, 10.13523668787394861766541840717, 11.24871498342065039707483561981
