| L(s) = 1 | + (−17.4 − 30.1i)2-s + (−351. + 609. i)4-s + (1.00e3 − 1.73e3i)5-s + (−2.43e3 − 4.21e3i)7-s + 6.68e3·8-s − 6.99e4·10-s + (−3.69e4 − 6.40e4i)11-s + (−1.32e4 + 2.29e4i)13-s + (−8.48e4 + 1.46e5i)14-s + (6.36e4 + 1.10e5i)16-s + 3.00e5·17-s + 1.80e5·19-s + (7.05e5 + 1.22e6i)20-s + (−1.28e6 + 2.23e6i)22-s + (−1.06e6 + 1.84e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.770 − 1.33i)2-s + (−0.687 + 1.19i)4-s + (0.717 − 1.24i)5-s + (−0.383 − 0.663i)7-s + 0.576·8-s − 2.21·10-s + (−0.761 − 1.31i)11-s + (−0.128 + 0.222i)13-s + (−0.590 + 1.02i)14-s + (0.242 + 0.420i)16-s + 0.872·17-s + 0.317·19-s + (0.986 + 1.70i)20-s + (−1.17 + 2.03i)22-s + (−0.794 + 1.37i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.414324 + 0.551020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414324 + 0.551020i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (17.4 + 30.1i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (-1.00e3 + 1.73e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (2.43e3 + 4.21e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.69e4 + 6.40e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.32e4 - 2.29e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 - 3.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.80e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (1.06e6 - 1.84e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (9.48e5 + 1.64e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (2.53e6 - 4.39e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 2.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (3.81e6 - 6.61e6i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-6.86e6 - 1.18e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (2.53e7 + 4.38e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + 2.60e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-2.11e7 + 3.65e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-2.43e7 - 4.22e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.56e7 + 1.48e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 8.48e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (9.50e7 + 1.64e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (2.21e8 + 3.84e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 4.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (8.06e8 + 1.39e9i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.73970337447111689262599985432, −12.91818332874678006447418441099, −11.64607361128334637562443235688, −10.27308212185709998913094758274, −9.366568329433775535842972866336, −8.145653973768756415600668861182, −5.55401411981011549405967188998, −3.41445794998918978177150225628, −1.53223247950095836764446305623, −0.36969126817551384287045293819,
2.53928645063913943426000490012, 5.55691999555602550489433048140, 6.69117727992518170263711720328, 7.79784863307226573134968163984, 9.515472832763790853293749084675, 10.34238203631191399860549582835, 12.48822701304613761793560522794, 14.31120820538867343226223307948, 15.03105528845339662429398321681, 16.06172615035816075769218718933
