| 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
−14.80685239276899511548331024132, −13.68800826595401983981322663565, −13.10711337584563633260428435902, −11.78257209086354092964682335854, −10.01084340645602085041012369178, −7.57635705002286207186391850460, −6.62834426322252734879023051398, −5.18616481619503608644155530288, −3.14590903350629512708394065617, −2.18905033640145226876205660549,
1.94451838794144457326998365663, 3.43243052536110127597877524235, 4.90055027159946987652626258473, 5.75049799553767186288829534712, 8.352362002826939571067098040811, 9.933540548464137382260445273488, 11.39043870749390587828251659973, 12.92876641687890205306227224066, 13.69802253478581709293062929433, 14.55279989970259576407116881396
