| 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 | \( 1 + (-7.17e4 + 6.01e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−16.68988409026364797424840514824, −15.92333811406181578536407326762, −14.74591984342529130076552651929, −13.78148127034198733901275318246, −11.88839315133104622691665415972, −10.31262121402878305510657015712, −8.861722126209150343418352704607, −6.84803692623348462588583237011, −5.90014861810763475337106964660, −4.24236125513819043833537511220,
0.836983105702390771135146287759, 2.75675486278850889703033446820, 5.17656129445870954518977304667, 7.11555903847716050315409842943, 9.387825351335057949734329048542, 10.76920411137447808310714224892, 11.87010177456047201943859381021, 12.87010421640150371785501693875, 13.56951924383340823388802501720, 15.69390649086353535208194048877
