| L(s) = 1 | + (−778. + 449. i)2-s + (1.95e4 − 2.14e3i)3-s + (2.73e5 − 4.72e5i)4-s + (−2.42e6 − 1.40e6i)5-s + (−1.42e7 + 1.04e7i)6-s + (−3.53e7 − 6.12e7i)7-s + 2.55e8i·8-s + (3.78e8 − 8.40e7i)9-s + 2.51e9·10-s + (−1.10e9 + 6.39e8i)11-s + (4.32e9 − 9.83e9i)12-s + (−1.60e9 + 2.78e9i)13-s + (5.50e10 + 3.17e10i)14-s + (−5.04e10 − 2.21e10i)15-s + (−4.31e10 − 7.47e10i)16-s + 1.18e11i·17-s + ⋯ |
| L(s) = 1 | + (−1.52 + 0.877i)2-s + (0.994 − 0.109i)3-s + (1.04 − 1.80i)4-s + (−1.24 − 0.716i)5-s + (−1.41 + 1.03i)6-s + (−0.876 − 1.51i)7-s + 1.90i·8-s + (0.976 − 0.216i)9-s + 2.51·10-s + (−0.469 + 0.271i)11-s + (0.838 − 1.90i)12-s + (−0.151 + 0.262i)13-s + (2.66 + 1.53i)14-s + (−1.31 − 0.577i)15-s + (−0.628 − 1.08i)16-s + 0.997i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.0605948 + 0.178625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0605948 + 0.178625i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.95e4 + 2.14e3i)T \) |
| good | 2 | \( 1 + (778. - 449. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (2.42e6 + 1.40e6i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (3.53e7 + 6.12e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (1.10e9 - 6.39e8i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (1.60e9 - 2.78e9i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 - 1.18e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 2.39e10T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-1.41e12 - 8.19e11i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (2.00e13 - 1.15e13i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (-4.75e12 + 8.22e12i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 - 1.93e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + (-1.09e14 - 6.32e13i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (1.34e14 + 2.32e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (-8.40e14 + 4.85e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 - 2.20e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (1.44e15 + 8.33e14i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (-3.55e15 - 6.15e15i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (2.01e16 - 3.49e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 + 4.09e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 1.72e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (5.39e16 + 9.33e16i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (2.16e17 - 1.25e17i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 - 2.35e16iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-6.45e17 - 1.11e18i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−17.01370879676847822260292273252, −16.09169442308596660335408837433, −15.01353721747074243785396083422, −13.03773107416459029016275252964, −10.48180280120817829317233349855, −9.130526759829292201757787707301, −7.82791001212085462740301199736, −7.08878025732601810265823303770, −3.90373421562488841968603980991, −1.13517505963505152129266332729,
0.11805121934167743943731683739, 2.50788294721899421275179295163, 3.21961570486059594278144788916, 7.36266565954302583098300745246, 8.565204655843905415857229376207, 9.666941054630048865568307641632, 11.26516459324021724745470799754, 12.58020928587384257637075389883, 15.19509784434298272342009156421, 16.08002486386019556196987265525
