| L(s) = 1 | + (−3.14 − 4.70i)2-s + (5.80 − 10.0i)3-s + (−12.1 + 29.5i)4-s + (−48.4 + 83.8i)5-s + (−65.5 + 4.36i)6-s − 77.0i·7-s + (177. − 35.8i)8-s + (54.1 + 93.7i)9-s + (546. − 36.3i)10-s − 612. i·11-s + (226. + 294. i)12-s + (784. − 453. i)13-s + (−362. + 242. i)14-s + (562. + 973. i)15-s + (−727. − 721. i)16-s + (−469. + 812. i)17-s + ⋯ |
| L(s) = 1 | + (−0.556 − 0.830i)2-s + (0.372 − 0.644i)3-s + (−0.380 + 0.924i)4-s + (−0.866 + 1.50i)5-s + (−0.742 + 0.0494i)6-s − 0.594i·7-s + (0.980 − 0.198i)8-s + (0.222 + 0.385i)9-s + (1.72 − 0.115i)10-s − 1.52i·11-s + (0.454 + 0.589i)12-s + (1.28 − 0.743i)13-s + (−0.493 + 0.330i)14-s + (0.644 + 1.11i)15-s + (−0.709 − 0.704i)16-s + (−0.393 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.01311 - 0.807056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01311 - 0.807056i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.14 + 4.70i)T \) |
| 19 | \( 1 + (-1.57e3 - 8.46i)T \) |
| good | 3 | \( 1 + (-5.80 + 10.0i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (48.4 - 83.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 77.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 612. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-784. + 453. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (469. - 812. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-156. + 90.1i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-5.13e3 + 2.96e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 4.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 573. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.74e3 - 1.58e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.17e4 + 6.80e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.77e3 - 1.02e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.46e4 + 8.44e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.73e4 - 3.01e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.67e3 - 1.15e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (907. + 1.57e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.84e4 + 6.66e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-3.05e4 + 5.28e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.40e4 + 5.90e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 9.86e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (9.28e4 - 5.35e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (4.25e4 + 2.45e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.56935060894265986355571021992, −11.89642554609744214007594977292, −10.86916954033149416155517117961, −10.46200483059697718341117859451, −8.381269487869407226695972453444, −7.80069613365853793940936954289, −6.56765062003738186798624822654, −3.74734812437913788133929897105, −2.86824765098888222579243126949, −0.860756510321295533280180839269,
1.15528693419179271104740102776, 4.20840612664635056446300221810, 5.04381123656599003548916014654, 6.88429559251694235322941850121, 8.315252329203778912198126596306, 9.065147853910924392713358593926, 9.794867832085228358725943493160, 11.61679082384614838433273277749, 12.66645173328315246986244947202, 14.03873351798507727554454784642
