| L(s) = 1 | + (1.93 − 1.62i)2-s + (0.766 − 4.34i)4-s + (−0.439 + 0.160i)5-s + (−0.560 − 3.17i)7-s + (−3.05 − 5.28i)8-s + (−0.592 + 1.02i)10-s + (2.91 + 1.06i)11-s + (−1.67 − 1.40i)13-s + (−6.25 − 5.25i)14-s + (−6.23 − 2.27i)16-s + (−1.5 + 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (0.358 + 2.03i)20-s + (7.39 − 2.68i)22-s + (1.06 − 6.01i)23-s + ⋯ |
| L(s) = 1 | + (1.37 − 1.15i)2-s + (0.383 − 2.17i)4-s + (−0.196 + 0.0715i)5-s + (−0.211 − 1.20i)7-s + (−1.07 − 1.86i)8-s + (−0.187 + 0.324i)10-s + (0.880 + 0.320i)11-s + (−0.464 − 0.389i)13-s + (−1.67 − 1.40i)14-s + (−1.55 − 0.567i)16-s + (−0.363 + 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (0.0801 + 0.454i)20-s + (1.57 − 0.573i)22-s + (0.221 − 1.25i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840696 - 2.80812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840696 - 2.80812i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.93 + 1.62i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (0.439 - 0.160i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.560 + 3.17i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.91 - 1.06i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.67 + 1.40i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0209 + 0.0362i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.06 + 6.01i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.03 + 4.22i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.08 - 6.13i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.79 - 3.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.90 - 4.95i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.553 - 0.201i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.67 + 9.51i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 + (-8.01 + 2.91i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.220 + 1.24i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.66 - 6.43i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.11 - 7.13i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.46 + 7.10i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.15 - 0.970i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.93 - 13.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (17.5 + 6.37i)T + (74.3 + 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29603560484927484464126955742, −9.770237709480656474336587313750, −8.385219386159557554462605217481, −7.03691717462193069411415895071, −6.37655434290125386680787726122, −5.12878695926888308230188449306, −4.20491600599441402437801394430, −3.67488832941401382268702382640, −2.43363313207530016194271628879, −1.04486583328236232143977769941,
2.44603266873717591675447308919, 3.58674815090456858342734524728, 4.53048941685112571740676081969, 5.49862282659755609221003639342, 6.14467404931489711732868236916, 7.00528455807814957186923484918, 7.84453208587462245190290281083, 8.875635813003625634906864026869, 9.505746085139234086394075896069, 11.20400303554619388475316287628
