| L(s) = 1 | + (0.595 + 0.499i)2-s + (−0.242 − 1.37i)4-s + (−2.23 − 0.812i)5-s + (−0.434 + 2.46i)7-s + (1.32 − 2.28i)8-s + (−0.924 − 1.60i)10-s + (−2.95 + 1.07i)11-s + (1.02 − 0.859i)13-s + (−1.49 + 1.25i)14-s + (−0.692 + 0.251i)16-s + (−3.13 − 5.43i)17-s + (−4.03 + 6.98i)19-s + (−0.575 + 3.26i)20-s + (−2.29 − 0.835i)22-s + (0.704 + 3.99i)23-s + ⋯ |
| L(s) = 1 | + (0.421 + 0.353i)2-s + (−0.121 − 0.686i)4-s + (−0.998 − 0.363i)5-s + (−0.164 + 0.931i)7-s + (0.466 − 0.808i)8-s + (−0.292 − 0.506i)10-s + (−0.890 + 0.323i)11-s + (0.283 − 0.238i)13-s + (−0.398 + 0.334i)14-s + (−0.173 + 0.0629i)16-s + (−0.760 − 1.31i)17-s + (−0.925 + 1.60i)19-s + (−0.128 + 0.730i)20-s + (−0.489 − 0.178i)22-s + (0.146 + 0.832i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00673395 - 0.115617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00673395 - 0.115617i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.595 - 0.499i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (2.23 + 0.812i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.434 - 2.46i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (2.95 - 1.07i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 0.859i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 - 6.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.704 - 3.99i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.11 + 5.96i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.491 - 2.78i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.76 + 4.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.44 - 4.56i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.19 - 0.799i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.801 - 4.54i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.135T + 53T^{2} \) |
| 59 | \( 1 + (-3.75 - 1.36i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0593 + 0.336i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.75 + 6.50i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.09 + 7.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.12 - 2.62i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.699 - 0.587i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.86 + 3.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.63 + 2.05i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.915562596679440491507407705422, −9.138366599436798249335591883098, −8.140621087368488020078950590786, −7.40007929709410930856508389603, −6.23059613862027373892426281859, −5.43985581130599532757212336484, −4.63842218884932587472420239988, −3.60468793845569681941598514682, −2.04307273419542690075035631298, −0.04890306003897926744243825155,
2.33608867077324156473918002256, 3.57260132738447369441952121538, 4.08076808873338469367408969683, 5.08045689603152716229771311394, 6.70741169273543293526442529620, 7.27509461514754255534744555295, 8.303702758085711633868678948037, 8.748698135143907436773756664962, 10.37881932600448020445448686926, 10.99419091786845343449991044022
