| L(s) = 1 | + (13.3 + 2.35i)2-s + (34.9 + 73.0i)3-s + (−68.2 − 24.8i)4-s + (−636. − 758. i)5-s + (293. + 1.05e3i)6-s + (−2.50e3 + 910. i)7-s + (−3.85e3 − 2.22e3i)8-s + (−4.12e3 + 5.10e3i)9-s + (−6.70e3 − 1.16e4i)10-s + (1.53e4 − 1.82e4i)11-s + (−568. − 5.85e3i)12-s + (4.78e3 + 2.71e4i)13-s + (−3.55e4 + 6.26e3i)14-s + (3.32e4 − 7.30e4i)15-s + (−3.19e4 − 2.67e4i)16-s + (−1.05e5 + 6.11e4i)17-s + ⋯ |
| L(s) = 1 | + (0.833 + 0.146i)2-s + (0.431 + 0.902i)3-s + (−0.266 − 0.0969i)4-s + (−1.01 − 1.21i)5-s + (0.226 + 0.815i)6-s + (−1.04 + 0.379i)7-s + (−0.940 − 0.543i)8-s + (−0.628 + 0.778i)9-s + (−0.670 − 1.16i)10-s + (1.04 − 1.24i)11-s + (−0.0273 − 0.282i)12-s + (0.167 + 0.949i)13-s + (−0.924 + 0.162i)14-s + (0.656 − 1.44i)15-s + (−0.487 − 0.408i)16-s + (−1.26 + 0.732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0694615 - 0.232251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0694615 - 0.232251i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-34.9 - 73.0i)T \) |
| good | 2 | \( 1 + (-13.3 - 2.35i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (636. + 758. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (2.50e3 - 910. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (-1.53e4 + 1.82e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (-4.78e3 - 2.71e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (1.05e5 - 6.11e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (547. - 948. i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-5.04e4 + 1.38e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (9.24e5 + 1.62e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-1.31e5 - 4.77e4i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (3.79e5 + 6.56e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-4.60e6 + 8.11e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (1.82e6 + 1.53e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-4.25e5 - 1.16e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 + 6.96e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.48e6 - 5.34e6i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (2.36e7 - 8.62e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (3.60e6 + 2.04e7i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (1.06e7 - 6.15e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-3.50e6 + 6.06e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.79e6 + 1.58e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-4.48e6 - 7.90e5i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (3.86e7 + 2.23e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.19e7 - 1.00e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−15.08254753140684322794731764602, −13.77126084988621914425806778353, −12.69827118388924086186739268016, −11.38986769882441849221413748297, −9.137690849532805286160528413434, −8.804847581428747357768743005045, −6.05038645102410975567341546906, −4.38766193496256331553256092507, −3.63706218245358341124769439393, −0.07807189504806760443302508006,
2.87168016431840652972255322084, 3.92571418122463189097156215659, 6.51220637851002245412949328410, 7.48775488311021542266022601897, 9.330559338538046626523958256180, 11.36084024282447950457254180656, 12.48280815019947323603455516998, 13.42482349615635644231092510855, 14.64647714271924704531211720295, 15.38111182864377364686208210500
