| L(s) = 1 | + (−0.898 − 5.09i)2-s + (6.18 − 46.3i)3-s + (95.1 − 34.6i)4-s + (331. + 278. i)5-s + (−241. + 10.1i)6-s + (981. + 357. i)7-s + (−592. − 1.02e3i)8-s + (−2.11e3 − 573. i)9-s + (1.11e3 − 1.93e3i)10-s + (2.11e3 − 1.77e3i)11-s + (−1.01e3 − 4.62e3i)12-s + (−193. + 1.09e3i)13-s + (937. − 5.31e3i)14-s + (1.49e4 − 1.36e4i)15-s + (5.23e3 − 4.38e3i)16-s + (−1.47e4 + 2.55e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0793 − 0.450i)2-s + (0.132 − 0.991i)3-s + (0.743 − 0.270i)4-s + (1.18 + 0.995i)5-s + (−0.456 + 0.0191i)6-s + (1.08 + 0.393i)7-s + (−0.409 − 0.709i)8-s + (−0.965 − 0.262i)9-s + (0.353 − 0.612i)10-s + (0.478 − 0.401i)11-s + (−0.169 − 0.772i)12-s + (−0.0243 + 0.138i)13-s + (0.0913 − 0.518i)14-s + (1.14 − 1.04i)15-s + (0.319 − 0.267i)16-s + (−0.729 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.02183 - 1.40167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02183 - 1.40167i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-6.18 + 46.3i)T \) |
| good | 2 | \( 1 + (0.898 + 5.09i)T + (-120. + 43.7i)T^{2} \) |
| 5 | \( 1 + (-331. - 278. i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-981. - 357. i)T + (6.30e5 + 5.29e5i)T^{2} \) |
| 11 | \( 1 + (-2.11e3 + 1.77e3i)T + (3.38e6 - 1.91e7i)T^{2} \) |
| 13 | \( 1 + (193. - 1.09e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.47e4 - 2.55e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.91e4 + 5.04e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.55e4 - 5.65e3i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-2.06e4 - 1.16e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (2.57e4 - 9.37e3i)T + (2.10e10 - 1.76e10i)T^{2} \) |
| 37 | \( 1 + (-4.71e4 + 8.16e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (9.35e4 - 5.30e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-6.30e5 + 5.29e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (1.89e5 + 6.90e4i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 - 1.53e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.08e5 - 3.42e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (2.36e6 + 8.61e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (5.93e5 - 3.36e6i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (2.10e6 - 3.64e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.45e5 - 4.25e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.13e5 - 4.61e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (1.46e6 + 8.31e6i)T + (-2.54e13 + 9.28e12i)T^{2} \) |
| 89 | \( 1 + (-1.21e6 - 2.10e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-9.94e6 + 8.34e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−15.07032300663352522559140343137, −14.30454970139647307799641021280, −12.98971634247139663539757259200, −11.44103757176939562976259683759, −10.70211240545356403240299146267, −8.830600944297351213844088335273, −6.90823799317586027794297243735, −6.00160896452469611920655050253, −2.54263849165993900247417722554, −1.62998822560010899730111601070,
1.98307729758263922791330421625, 4.57487420377457250847905210787, 5.94444984447702760715794709590, 8.044836940995963121912225210028, 9.296242394565610900834063043176, 10.68059620599195481872193352778, 12.06376267721664457901473216277, 13.88375927737078182121991674792, 14.85253910074950409430870533640, 16.20966641484229421624334533898
