| L(s) = 1 | + (437. − 252. i)2-s + (−7.96e3 + 1.80e4i)3-s + (−3.38e3 + 5.86e3i)4-s + (−6.93e5 − 4.00e5i)5-s + (1.06e6 + 9.88e6i)6-s + (−2.35e7 − 4.07e7i)7-s + 1.35e8i·8-s + (−2.60e8 − 2.86e8i)9-s − 4.04e8·10-s + (3.96e9 − 2.28e9i)11-s + (−7.86e7 − 1.07e8i)12-s + (7.58e9 − 1.31e10i)13-s + (−2.06e10 − 1.19e10i)14-s + (1.27e10 − 9.29e9i)15-s + (3.34e10 + 5.79e10i)16-s − 1.64e10i·17-s + ⋯ |
| L(s) = 1 | + (0.854 − 0.493i)2-s + (−0.404 + 0.914i)3-s + (−0.0129 + 0.0223i)4-s + (−0.355 − 0.204i)5-s + (0.105 + 0.981i)6-s + (−0.583 − 1.01i)7-s + 1.01i·8-s + (−0.672 − 0.739i)9-s − 0.404·10-s + (1.67 − 0.969i)11-s + (−0.0152 − 0.0208i)12-s + (0.715 − 1.23i)13-s + (−0.997 − 0.576i)14-s + (0.331 − 0.241i)15-s + (0.486 + 0.843i)16-s − 0.138i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.01796 - 1.16515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01796 - 1.16515i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.96e3 - 1.80e4i)T \) |
| good | 2 | \( 1 + (-437. + 252. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (6.93e5 + 4.00e5i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (2.35e7 + 4.07e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (-3.96e9 + 2.28e9i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (-7.58e9 + 1.31e10i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 + 1.64e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 4.16e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (6.08e11 + 3.51e11i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (1.62e13 - 9.39e12i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-1.04e13 + 1.81e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 - 1.49e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + (1.26e14 + 7.31e13i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (-1.73e13 - 3.00e13i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (-7.42e14 + 4.28e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 + 3.44e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (-1.12e15 - 6.49e14i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (-6.70e15 - 1.16e16i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (1.59e16 - 2.76e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 + 5.68e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 2.38e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (9.02e16 + 1.56e17i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (-5.58e16 + 3.22e16i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 + 1.36e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-5.79e17 - 1.00e18i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−16.39515551318854760453999963753, −14.66521541821328752571671771882, −13.28230271695235599964574940943, −11.76014119824844929823337481515, −10.58182810764097175190335202638, −8.643972424217017736860066544022, −6.03376956301892290704597798091, −4.12611937125946223927831949063, −3.51118519863660946294977416216, −0.47335055901071461772801475551,
1.68543494031466022542692437700, 4.09350176568590751147196066791, 6.04568917729942167282336249623, 6.84005783607893448891125279598, 9.179638083368271659836099373202, 11.68255003859727157108185934546, 12.72680539418027495145998188062, 14.17116190630665233329112658093, 15.37123225761118964370165956992, 16.97162005203060229509727042064
