| L(s) = 1 | + (1.40 − 7.95i)2-s + (−9.89 − 12.0i)3-s + (−31.2 − 11.3i)4-s + (15.2 − 12.7i)5-s + (−109. + 61.8i)6-s + (−70.6 + 25.7i)7-s + (−5.20 + 9.00i)8-s + (−47.1 + 238. i)9-s + (−80.3 − 139. i)10-s + (144. + 121. i)11-s + (172. + 489. i)12-s + (−145. − 824. i)13-s + (105. + 598. i)14-s + (−304. − 56.9i)15-s + (−751. − 630. i)16-s + (−465. − 805. i)17-s + ⋯ |
| L(s) = 1 | + (0.248 − 1.40i)2-s + (−0.634 − 0.772i)3-s + (−0.977 − 0.355i)4-s + (0.272 − 0.228i)5-s + (−1.24 + 0.701i)6-s + (−0.545 + 0.198i)7-s + (−0.0287 + 0.0497i)8-s + (−0.194 + 0.980i)9-s + (−0.253 − 0.439i)10-s + (0.360 + 0.302i)11-s + (0.345 + 0.981i)12-s + (−0.238 − 1.35i)13-s + (0.143 + 0.816i)14-s + (−0.349 − 0.0653i)15-s + (−0.733 − 0.615i)16-s + (−0.390 − 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.159772 + 1.20341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159772 + 1.20341i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (9.89 + 12.0i)T \) |
| good | 2 | \( 1 + (-1.40 + 7.95i)T + (-30.0 - 10.9i)T^{2} \) |
| 5 | \( 1 + (-15.2 + 12.7i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (70.6 - 25.7i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-144. - 121. i)T + (2.79e4 + 1.58e5i)T^{2} \) |
| 13 | \( 1 + (145. + 824. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (465. + 805. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.07e3 + 1.86e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.22e3 - 809. i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (17.6 - 100. i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-5.47e3 - 1.99e3i)T + (2.19e7 + 1.84e7i)T^{2} \) |
| 37 | \( 1 + (5.39e3 + 9.33e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-2.49e3 - 1.41e4i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-512. - 430. i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-2.67e4 + 9.72e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 - 2.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.17e4 + 9.86e3i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (8.42e3 - 3.06e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (1.06e4 + 6.03e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (2.55e3 + 4.42e3i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.76e4 - 3.06e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (9.55e3 - 5.41e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (1.65e4 - 9.41e4i)T + (-3.70e9 - 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-5.03e4 + 8.72e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-7.17e4 - 6.01e4i)T + (1.49e9 + 8.45e9i)T^{2} \) |
| 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
−15.69390649086353535208194048877, −13.56951924383340823388802501720, −12.87010421640150371785501693875, −11.87010177456047201943859381021, −10.76920411137447808310714224892, −9.387825351335057949734329048542, −7.11555903847716050315409842943, −5.17656129445870954518977304667, −2.75675486278850889703033446820, −0.836983105702390771135146287759,
4.24236125513819043833537511220, 5.90014861810763475337106964660, 6.84803692623348462588583237011, 8.861722126209150343418352704607, 10.31262121402878305510657015712, 11.88839315133104622691665415972, 13.78148127034198733901275318246, 14.74591984342529130076552651929, 15.92333811406181578536407326762, 16.68988409026364797424840514824
