| L(s) = 1 | + (26.1 − 31.1i)2-s + (242. − 6.50i)3-s + (−110. − 623. i)4-s + (−1.91e3 + 5.24e3i)5-s + (6.15e3 − 7.74e3i)6-s + (1.31e3 − 7.47e3i)7-s + (1.37e4 + 7.94e3i)8-s + (5.89e4 − 3.16e3i)9-s + (1.13e5 + 1.96e5i)10-s + (5.83e4 + 1.60e5i)11-s + (−3.07e4 − 1.50e5i)12-s + (5.42e5 − 4.55e5i)13-s + (−1.98e5 − 2.36e5i)14-s + (−4.30e5 + 1.28e6i)15-s + (1.21e6 − 4.43e5i)16-s + (−1.73e6 + 1.00e6i)17-s + ⋯ |
| L(s) = 1 | + (0.817 − 0.974i)2-s + (0.999 − 0.0267i)3-s + (−0.107 − 0.609i)4-s + (−0.611 + 1.67i)5-s + (0.791 − 0.996i)6-s + (0.0784 − 0.444i)7-s + (0.420 + 0.242i)8-s + (0.998 − 0.0535i)9-s + (1.13 + 1.96i)10-s + (0.362 + 0.995i)11-s + (−0.123 − 0.606i)12-s + (1.46 − 1.22i)13-s + (−0.369 − 0.440i)14-s + (−0.566 + 1.69i)15-s + (1.16 − 0.422i)16-s + (−1.22 + 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.98145 - 0.303966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.98145 - 0.303966i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-242. + 6.50i)T \) |
| good | 2 | \( 1 + (-26.1 + 31.1i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (1.91e3 - 5.24e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (-1.31e3 + 7.47e3i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-5.83e4 - 1.60e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-5.42e5 + 4.55e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (1.73e6 - 1.00e6i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (1.02e6 - 1.76e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-4.79e6 + 8.45e5i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (2.44e5 - 2.91e5i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (-9.47e5 - 5.37e6i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (4.95e7 + 8.57e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (6.27e7 + 7.47e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (1.79e8 - 6.53e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (7.21e7 + 1.27e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 - 7.34e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.41e8 + 6.62e8i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-1.13e8 + 6.40e8i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-7.57e8 + 6.35e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (8.90e8 - 5.13e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-7.35e8 + 1.27e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-1.00e9 - 8.41e8i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-2.30e8 + 2.75e8i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-1.03e9 - 6.00e8i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (2.58e9 - 9.40e8i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−14.74062182055522575976531036307, −13.68323259187285842020148789973, −12.62402308440894625031367411218, −10.97344059713083041662277233562, −10.39317694817460852275719385647, −8.129358931265146081891077874291, −6.84707809311430173902185666586, −4.01216435648047654104409441100, −3.30123470643838103069642844179, −1.93546689018598918128502919025,
1.26439966920698500869746736958, 3.89492419537251967795349837013, 4.90418126892116859041039799776, 6.68929958630398185474449827949, 8.498370557704732174487852038535, 8.934512429608950588602068617981, 11.57979036869792818509109023625, 13.30306357382564275385807601488, 13.57748882637814188523844616554, 15.23000980555232262565898695723
