| L(s) = 1 | + (166. − 287. i)2-s + (−3.88e4 − 6.72e4i)4-s + (−9.17e4 − 1.58e5i)5-s + (1.67e6 − 2.90e6i)7-s − 1.49e7·8-s − 6.09e7·10-s + (1.53e7 − 2.66e7i)11-s + (−1.11e8 − 1.93e8i)13-s + (−5.57e8 − 9.65e8i)14-s + (−1.20e9 + 2.08e9i)16-s + 4.16e8·17-s + 3.21e9·19-s + (−7.12e9 + 1.23e10i)20-s + (−5.11e9 − 8.85e9i)22-s + (1.10e10 + 1.90e10i)23-s + ⋯ |
| L(s) = 1 | + (0.917 − 1.58i)2-s + (−1.18 − 2.05i)4-s + (−0.525 − 0.909i)5-s + (0.770 − 1.33i)7-s − 2.51·8-s − 1.92·10-s + (0.238 − 0.412i)11-s + (−0.493 − 0.855i)13-s + (−1.41 − 2.44i)14-s + (−1.12 + 1.94i)16-s + 0.246·17-s + 0.825·19-s + (−1.24 + 2.15i)20-s + (−0.437 − 0.757i)22-s + (0.673 + 1.16i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.766127736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766127736\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (-166. + 287. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (9.17e4 + 1.58e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-1.67e6 + 2.90e6i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-1.53e7 + 2.66e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (1.11e8 + 1.93e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 - 4.16e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.21e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-1.10e10 - 1.90e10i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (2.44e10 - 4.22e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (-4.52e9 - 7.83e9i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 - 8.40e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (-7.72e11 - 1.33e12i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (6.73e11 - 1.16e12i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-1.65e12 + 2.86e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + 1.35e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (-1.02e12 - 1.78e12i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-4.30e12 + 7.45e12i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (2.32e13 + 4.03e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 4.49e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.02e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-1.58e14 + 2.74e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (1.02e14 - 1.78e14i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 4.24e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (3.39e14 - 5.87e14i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
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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
−12.87992378705984058683682178453, −11.70789296538897151125709130524, −10.84614221254951987474409911498, −9.576961174028846422797694573174, −7.81179903051696264136593716480, −5.27104107677452831866930887648, −4.35427237646362956732060306325, −3.22376574470118906565153356687, −1.28248046597105113123837975677, −0.66989739049004113974714317577,
2.65216217625559318567338793843, 4.27746923329401851100692064985, 5.47544264026143140025046612311, 6.77222571858237094585439960692, 7.77288204007514456859270078678, 9.066421025371344055421833159643, 11.51147116965976754459424518398, 12.49471246125519818128884936796, 14.19960254260444680901471057262, 14.82269408447850634478962042351
