| L(s) = 1 | + (−137. + 238. i)2-s + (−2.15e4 − 3.73e4i)4-s + (−1.32e5 − 2.30e5i)5-s + (1.52e6 − 2.64e6i)7-s + 2.84e6·8-s + 7.31e7·10-s + (3.23e7 − 5.61e7i)11-s + (−1.10e8 − 1.92e8i)13-s + (4.20e8 + 7.27e8i)14-s + (3.13e8 − 5.43e8i)16-s − 3.44e8·17-s − 5.53e9·19-s + (−5.72e9 + 9.91e9i)20-s + (8.92e9 + 1.54e10i)22-s + (2.61e9 + 4.53e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.760 + 1.31i)2-s + (−0.657 − 1.13i)4-s + (−0.760 − 1.31i)5-s + (0.699 − 1.21i)7-s + 0.480·8-s + 2.31·10-s + (0.501 − 0.868i)11-s + (−0.490 − 0.849i)13-s + (1.06 + 1.84i)14-s + (0.292 − 0.506i)16-s − 0.203·17-s − 1.41·19-s + (−1.00 + 1.73i)20-s + (0.762 + 1.32i)22-s + (0.160 + 0.277i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.4647905896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4647905896\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (137. - 238. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (1.32e5 + 2.30e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-1.52e6 + 2.64e6i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-3.23e7 + 5.61e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (1.10e8 + 1.92e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 + 3.44e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 5.53e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-2.61e9 - 4.53e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (-1.31e10 + 2.27e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (-6.50e10 - 1.12e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + 4.21e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (6.25e11 + 1.08e12i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-1.13e12 + 1.96e12i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-7.66e11 + 1.32e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 - 9.47e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (-1.97e12 - 3.41e12i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (1.21e13 - 2.10e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-1.82e13 - 3.16e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 1.25e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.07e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-5.92e13 + 1.02e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (5.33e13 - 9.24e13i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 2.96e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (-1.66e14 + 2.88e14i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
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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
−13.69152447162021309508962736466, −12.16818265477249183245691411473, −10.54527713715663158585670745465, −8.784631103155042160159092015544, −8.166410241074969166860226943550, −7.04668329802067815550746634447, −5.34612609660286068565133381604, −4.07123989720488366190060072819, −0.974662946878188495651333427806, −0.21734223103567053349584467401,
1.88019923102087735259585996680, 2.67562189012934737474733492127, 4.24863415342202466724872289103, 6.63370289195011102097120559408, 8.233955205535367660409140341905, 9.460860172898110778632466062646, 10.79798159855401551432302669547, 11.62337456556269043764167062387, 12.38669730439730919982649189070, 14.64336525970023435159859285868
