| L(s) = 1 | + (14.6 + 12.2i)2-s + (−8.64 + 45.9i)3-s + (40.9 + 232. i)4-s + (357. + 130. i)5-s + (−689. + 565. i)6-s + (177. − 1.00e3i)7-s + (−1.02e3 + 1.77e3i)8-s + (−2.03e3 − 794. i)9-s + (3.62e3 + 6.27e3i)10-s + (−5.22e3 + 1.90e3i)11-s + (−1.10e4 − 125. i)12-s + (7.02e3 − 5.89e3i)13-s + (1.49e4 − 1.25e4i)14-s + (−9.06e3 + 1.52e4i)15-s + (−8.45e3 + 3.07e3i)16-s + (801. + 1.38e3i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 1.08i)2-s + (−0.184 + 0.982i)3-s + (0.319 + 1.81i)4-s + (1.27 + 0.465i)5-s + (−1.30 + 1.06i)6-s + (0.195 − 1.11i)7-s + (−0.709 + 1.22i)8-s + (−0.931 − 0.363i)9-s + (1.14 + 1.98i)10-s + (−1.18 + 0.430i)11-s + (−1.84 − 0.0208i)12-s + (0.887 − 0.744i)13-s + (1.45 − 1.22i)14-s + (−0.693 + 1.17i)15-s + (−0.515 + 0.187i)16-s + (0.0395 + 0.0685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.35857 + 3.25708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35857 + 3.25708i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.64 - 45.9i)T \) |
| good | 2 | \( 1 + (-14.6 - 12.2i)T + (22.2 + 126. i)T^{2} \) |
| 5 | \( 1 + (-357. - 130. i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-177. + 1.00e3i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (5.22e3 - 1.90e3i)T + (1.49e7 - 1.25e7i)T^{2} \) |
| 13 | \( 1 + (-7.02e3 + 5.89e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-801. - 1.38e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.53e4 + 2.65e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.87e4 - 1.06e5i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (1.22e5 + 1.03e5i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-1.23e3 - 7.02e3i)T + (-2.58e10 + 9.40e9i)T^{2} \) |
| 37 | \( 1 + (-1.59e4 - 2.76e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.33e3 + 1.95e3i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (3.78e5 - 1.37e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (4.41e4 - 2.50e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 - 2.41e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.77e6 + 6.45e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-5.53e5 + 3.13e6i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.42e6 - 1.19e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (2.08e6 + 3.60e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.18e5 - 2.05e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (5.22e6 + 4.38e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-7.40e6 - 6.21e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (-1.12e6 + 1.95e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.10e6 + 1.49e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−15.84318758106661593200913890378, −15.05781011776727992595476203911, −13.72497334622914044785264549797, −13.30940952250493931687543451932, −10.99839266500062226285688303675, −9.834290490819120406890710318899, −7.57174535351941570184874621131, −5.97295443814600881895098972237, −4.99028573165609549437066682901, −3.33163474454408432018832304285,
1.55892129018318974862240275134, 2.61522977741815826375890191442, 5.29791433009801759835026923029, 6.01911396356348360223066500747, 8.714291434142891636832345897917, 10.58850359143399786755749093766, 11.88487946661422825254195152621, 12.87694331005243263852675889162, 13.55574897227129192512511452682, 14.57490438335021022276558816010
