| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (180. − 104. i)5-s + (−2.09 + 3.63i)7-s + 181. i·8-s + 1.17e3·10-s + (1.95e3 + 1.13e3i)11-s + (−1.42e3 − 2.45e3i)13-s + (−20.5 + 11.8i)14-s + (−512. + 886. i)16-s + 1.96e3i·17-s − 281.·19-s + (5.77e3 + 3.33e3i)20-s + (6.39e3 + 1.10e4i)22-s + (1.45e4 − 8.37e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.44 − 0.834i)5-s + (−0.00611 + 0.0105i)7-s + 0.353i·8-s + 1.17·10-s + (1.47 + 0.849i)11-s + (−0.646 − 1.11i)13-s + (−0.00749 + 0.00432i)14-s + (−0.125 + 0.216i)16-s + 0.400i·17-s − 0.0410·19-s + (0.722 + 0.417i)20-s + (0.600 + 1.04i)22-s + (1.19 − 0.688i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.144397785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.144397785\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-180. + 104. i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (2.09 - 3.63i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.95e3 - 1.13e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.42e3 + 2.45e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 1.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 281.T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.45e4 + 8.37e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (3.21e4 + 1.85e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.23e4 - 2.13e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 1.70e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.00e5 + 5.81e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.53e4 + 2.65e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.72e4 - 3.88e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.32e5 - 7.64e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-8.06e3 + 1.39e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.37e5 - 4.11e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.50e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.31e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-4.48e5 + 7.76e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (8.18e5 + 4.72e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 7.90e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (6.96e5 - 1.20e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.32655160051307051313126627289, −10.70356160916529237267450467417, −9.560736258767790082573087378340, −8.866368955862572969773262757511, −7.35140603724078099725013795391, −6.18889200898397756053415332138, −5.32017911953237524361478373810, −4.24596428752086627774621266271, −2.45796904954714194757540929572, −1.17221197598771822830256787170,
1.34918817098490835607613431479, 2.47406291476136074134517621688, 3.74168090690689966546753062254, 5.29689202147995012176860862778, 6.34405381032652807383559859410, 7.03752381753836561064390220696, 9.211880593775507871573084031520, 9.607669993870301013455313324453, 10.99600792435384224087438318704, 11.54710672264120084873501247525
