| L(s) = 1 | + (4.95 + 15.2i)2-s + (−21.8 − 15.8i)3-s + (−104. + 75.8i)4-s + (99.6 − 306. i)5-s + (133. − 411. i)6-s + (803. − 583. i)7-s + (−12.6 − 9.22i)8-s + (225. + 693. i)9-s + 5.17e3·10-s + (2.10e3 + 3.88e3i)11-s + 3.48e3·12-s + (−4.46e3 − 1.37e4i)13-s + (1.28e4 + 9.35e3i)14-s + (−7.04e3 + 5.12e3i)15-s + (−5.02e3 + 1.54e4i)16-s + (6.56e3 − 2.02e4i)17-s + ⋯ |
| L(s) = 1 | + (0.437 + 1.34i)2-s + (−0.467 − 0.339i)3-s + (−0.815 + 0.592i)4-s + (0.356 − 1.09i)5-s + (0.252 − 0.778i)6-s + (0.885 − 0.643i)7-s + (−0.00876 − 0.00636i)8-s + (0.103 + 0.317i)9-s + 1.63·10-s + (0.476 + 0.879i)11-s + 0.581·12-s + (−0.563 − 1.73i)13-s + (1.25 + 0.911i)14-s + (−0.539 + 0.391i)15-s + (−0.306 + 0.943i)16-s + (0.324 − 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.15256 + 0.494004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15256 + 0.494004i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (21.8 + 15.8i)T \) |
| 11 | \( 1 + (-2.10e3 - 3.88e3i)T \) |
| good | 2 | \( 1 + (-4.95 - 15.2i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (-99.6 + 306. i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-803. + 583. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (4.46e3 + 1.37e4i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-6.56e3 + 2.02e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-3.54e4 - 2.57e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 2.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.45e5 + 1.05e5i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (4.11e4 + 1.26e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.98e5 - 2.16e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (1.62e4 + 1.17e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 3.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (4.70e5 + 3.42e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-3.73e3 - 1.14e4i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-5.14e5 + 3.73e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (9.50e5 - 2.92e6i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 - 4.42e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (4.44e5 - 1.36e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (2.82e6 - 2.05e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-2.49e6 - 7.69e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-8.27e5 + 2.54e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 - 2.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.90e6 + 8.95e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.31524455052942694900403122979, −14.15021176239207365816002416389, −13.12705414469513918293943195077, −11.91624415137965388185823647658, −10.03598281177705498340326623163, −8.124349371621896715981430175866, −7.24915772432853463415266749406, −5.47200629820102761310390155988, −4.77876443184743279006728918896, −1.12942205700781337747178744576,
1.68247004367861664715824784301, 3.23807745292517818003812839807, 4.92371423256630705011253007432, 6.74463147552538460369044205246, 9.148542617227575086797847844495, 10.55313365279424700202372071310, 11.38092472964935224240317610109, 12.10192134558284030334902653179, 13.92762806621436186910135358253, 14.58144657977289454605336006232
