| L(s) = 1 | + (3.07 + 3.85i)2-s + (78.3 + 11.8i)3-s + (23.0 − 101. i)4-s + (247. + 168. i)5-s + (195. + 338. i)6-s + (−868. + 1.50e3i)7-s + (1.02e3 − 495. i)8-s + (3.90e3 + 1.20e3i)9-s + (110. + 1.47e3i)10-s + (−1.09e3 − 4.81e3i)11-s + (3.00e3 − 7.64e3i)12-s + (−229. + 3.05e3i)13-s + (−8.46e3 + 1.27e3i)14-s + (1.73e4 + 1.61e4i)15-s + (−6.87e3 − 3.31e3i)16-s + (1.57e4 − 1.07e4i)17-s + ⋯ |
| L(s) = 1 | + (0.271 + 0.340i)2-s + (1.67 + 0.252i)3-s + (0.180 − 0.789i)4-s + (0.885 + 0.603i)5-s + (0.369 + 0.639i)6-s + (−0.956 + 1.65i)7-s + (0.710 − 0.342i)8-s + (1.78 + 0.551i)9-s + (0.0349 + 0.465i)10-s + (−0.248 − 1.08i)11-s + (0.501 − 1.27i)12-s + (−0.0289 + 0.385i)13-s + (−0.824 + 0.124i)14-s + (1.33 + 1.23i)15-s + (−0.419 − 0.202i)16-s + (0.779 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.66972 + 1.39954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.66972 + 1.39954i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.73e5 - 3.64e5i)T \) |
| good | 2 | \( 1 + (-3.07 - 3.85i)T + (-28.4 + 124. i)T^{2} \) |
| 3 | \( 1 + (-78.3 - 11.8i)T + (2.08e3 + 644. i)T^{2} \) |
| 5 | \( 1 + (-247. - 168. i)T + (2.85e4 + 7.27e4i)T^{2} \) |
| 7 | \( 1 + (868. - 1.50e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.09e3 + 4.81e3i)T + (-1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (229. - 3.05e3i)T + (-6.20e7 - 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-1.57e4 + 1.07e4i)T + (1.49e8 - 3.81e8i)T^{2} \) |
| 19 | \( 1 + (2.17e4 - 6.69e3i)T + (7.38e8 - 5.03e8i)T^{2} \) |
| 23 | \( 1 + (-4.02e4 + 3.73e4i)T + (2.54e8 - 3.39e9i)T^{2} \) |
| 29 | \( 1 + (-7.02e4 + 1.05e4i)T + (1.64e10 - 5.08e9i)T^{2} \) |
| 31 | \( 1 + (1.05e5 - 2.68e5i)T + (-2.01e10 - 1.87e10i)T^{2} \) |
| 37 | \( 1 + (2.08e5 + 3.61e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (2.62e5 + 3.29e5i)T + (-4.33e10 + 1.89e11i)T^{2} \) |
| 47 | \( 1 + (-8.82e4 + 3.86e5i)T + (-4.56e11 - 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-5.00e4 - 6.68e5i)T + (-1.16e12 + 1.75e11i)T^{2} \) |
| 59 | \( 1 + (1.28e6 + 6.20e5i)T + (1.55e12 + 1.94e12i)T^{2} \) |
| 61 | \( 1 + (5.89e5 + 1.50e6i)T + (-2.30e12 + 2.13e12i)T^{2} \) |
| 67 | \( 1 + (-1.68e6 + 5.18e5i)T + (5.00e12 - 3.41e12i)T^{2} \) |
| 71 | \( 1 + (-1.20e6 - 1.11e6i)T + (6.79e11 + 9.06e12i)T^{2} \) |
| 73 | \( 1 + (2.21e5 - 2.95e6i)T + (-1.09e13 - 1.64e12i)T^{2} \) |
| 79 | \( 1 + (8.90e5 - 1.54e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (5.71e6 + 8.60e5i)T + (2.59e13 + 7.99e12i)T^{2} \) |
| 89 | \( 1 + (6.83e5 + 1.03e5i)T + (4.22e13 + 1.30e13i)T^{2} \) |
| 97 | \( 1 + (1.41e6 + 6.18e6i)T + (-7.27e13 + 3.50e13i)T^{2} \) |
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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
−14.38738511269637512935719360772, −14.00278096030830025979544426504, −12.69677322538642415130086521244, −10.51261204167679638026190089199, −9.462066667210601380735049709748, −8.673148007165411713177705389146, −6.69202297600361007637353697281, −5.53691241022301680616354286326, −3.10439668842813451082619903866, −2.16219384148076933620551590165,
1.62100664216380592609914089687, 3.08105655058346635987258172031, 4.21743945914748919884783192750, 7.05175651755455828745846866763, 7.940188827904186175667876731215, 9.412492126145259192648484762623, 10.31143018106295405228224119130, 12.71358660330909217134251141793, 13.17654113737510577840613800800, 13.82013017146121742250201839009
