| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−43.4 + 25.0i)5-s + (−28.1 + 48.7i)7-s − 181. i·8-s + 283.·10-s + (1.41e3 + 815. i)11-s + (−1.73e3 − 3.00e3i)13-s + (275. − 159. i)14-s + (−512. + 886. i)16-s + 599. i·17-s + 5.35e3·19-s + (−1.38e3 − 802. i)20-s + (−4.61e3 − 7.99e3i)22-s + (817. − 472. i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.347 + 0.200i)5-s + (−0.0820 + 0.142i)7-s − 0.353i·8-s + 0.283·10-s + (1.06 + 0.612i)11-s + (−0.789 − 1.36i)13-s + (0.100 − 0.0579i)14-s + (−0.125 + 0.216i)16-s + 0.122i·17-s + 0.780·19-s + (−0.173 − 0.100i)20-s + (−0.433 − 0.750i)22-s + (0.0672 − 0.0388i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8927599191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8927599191\) |
| \(L(4)\) |
|
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 | \( 1 + (43.4 - 25.0i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (28.1 - 48.7i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.41e3 - 815. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.73e3 + 3.00e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 599. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.35e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-817. + 472. i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.31e4 - 7.59e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.11e4 + 3.66e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 5.38e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (9.75e4 - 5.63e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.86e4 + 6.69e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.64e5 - 9.49e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.43e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (7.29e4 - 4.21e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (4.91e4 - 8.52e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.23e5 - 2.14e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.11e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.68e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (4.21e5 - 7.30e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.07e5 - 2.35e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 7.52e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (2.30e5 - 3.99e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.98648433737782388195193159432, −10.90245802600444172224688858375, −9.909935066072279758924961986440, −9.081705276222076940819172252400, −7.79919220660920692395204493725, −7.04406023341669772623898658576, −5.52580108505166168770314242204, −3.94144317848805113516100406979, −2.69861920301645846882080304269, −1.13954202397495634875074667169,
0.36314901612746936616025704562, 1.80126504033232866667128070529, 3.65886327391735437265839266736, 5.01820197349253609347048784773, 6.49542761086335769640872905446, 7.26609110833095919578585929744, 8.582674172882176895109032130528, 9.291667047579569528876727305875, 10.35991306476405106203884756482, 11.65639667478952452929092600178
