| L(s) = 1 | + (53.5 − 9.43i)2-s + (161. + 181. i)3-s + (1.81e3 − 659. i)4-s + (2.58e3 − 3.08e3i)5-s + (1.03e4 + 8.18e3i)6-s + (3.64e3 + 1.32e3i)7-s + (4.25e4 − 2.45e4i)8-s + (−6.77e3 + 5.86e4i)9-s + (1.09e5 − 1.89e5i)10-s + (−1.55e5 − 1.84e5i)11-s + (4.12e5 + 2.22e5i)12-s + (−4.02e4 + 2.28e5i)13-s + (2.07e5 + 3.66e4i)14-s + (9.77e5 − 2.90e4i)15-s + (5.33e5 − 4.47e5i)16-s + (5.05e5 + 2.91e5i)17-s + ⋯ |
| L(s) = 1 | + (1.67 − 0.294i)2-s + (0.665 + 0.746i)3-s + (1.76 − 0.644i)4-s + (0.827 − 0.986i)5-s + (1.33 + 1.05i)6-s + (0.216 + 0.0789i)7-s + (1.29 − 0.750i)8-s + (−0.114 + 0.993i)9-s + (1.09 − 1.89i)10-s + (−0.963 − 1.14i)11-s + (1.65 + 0.892i)12-s + (−0.108 + 0.614i)13-s + (0.386 + 0.0680i)14-s + (1.28 − 0.0383i)15-s + (0.508 − 0.426i)16-s + (0.356 + 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(6.32403 - 0.639628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.32403 - 0.639628i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-161. - 181. i)T \) |
| good | 2 | \( 1 + (-53.5 + 9.43i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (-2.58e3 + 3.08e3i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (-3.64e3 - 1.32e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (1.55e5 + 1.84e5i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (4.02e4 - 2.28e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (-5.05e5 - 2.91e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-1.36e6 - 2.37e6i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (9.32e5 + 2.56e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (3.08e7 - 5.44e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-4.34e7 + 1.58e7i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (-1.34e7 + 2.32e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (2.24e8 + 3.95e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (8.76e6 - 7.35e6i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (1.23e8 - 3.38e8i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 + 3.56e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-4.20e8 + 5.01e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (-8.91e8 - 3.24e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-2.11e8 + 1.19e9i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (7.84e8 + 4.52e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.96e9 - 3.40e9i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.44e8 + 8.16e8i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (3.09e9 - 5.46e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (-8.48e9 + 4.90e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-7.33e9 + 6.15e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−14.55279989970259576407116881396, −13.69802253478581709293062929433, −12.92876641687890205306227224066, −11.39043870749390587828251659973, −9.933540548464137382260445273488, −8.352362002826939571067098040811, −5.75049799553767186288829534712, −4.90055027159946987652626258473, −3.43243052536110127597877524235, −1.94451838794144457326998365663,
2.18905033640145226876205660549, 3.14590903350629512708394065617, 5.18616481619503608644155530288, 6.62834426322252734879023051398, 7.57635705002286207186391850460, 10.01084340645602085041012369178, 11.78257209086354092964682335854, 13.10711337584563633260428435902, 13.68800826595401983981322663565, 14.80685239276899511548331024132
