| L(s) = 1 | + (0.250 + 0.127i)2-s + (−1.71 − 0.270i)3-s + (−2.30 − 3.17i)4-s + (−4.47 − 2.22i)5-s + (−0.393 − 0.285i)6-s + (−6.72 + 6.72i)7-s + (−0.347 − 2.19i)8-s + (2.85 + 0.927i)9-s + (−0.836 − 1.12i)10-s + (−5.01 − 15.4i)11-s + (3.08 + 6.05i)12-s + (18.1 − 9.23i)13-s + (−2.53 + 0.824i)14-s + (7.05 + 5.01i)15-s + (−4.65 + 14.3i)16-s + (−9.63 + 1.52i)17-s + ⋯ |
| L(s) = 1 | + (0.125 + 0.0637i)2-s + (−0.570 − 0.0903i)3-s + (−0.576 − 0.793i)4-s + (−0.895 − 0.444i)5-s + (−0.0655 − 0.0476i)6-s + (−0.960 + 0.960i)7-s + (−0.0434 − 0.274i)8-s + (0.317 + 0.103i)9-s + (−0.0836 − 0.112i)10-s + (−0.456 − 1.40i)11-s + (0.256 + 0.504i)12-s + (1.39 − 0.710i)13-s + (−0.181 + 0.0589i)14-s + (0.470 + 0.334i)15-s + (−0.290 + 0.895i)16-s + (−0.566 + 0.0897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.114365 - 0.401914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.114365 - 0.401914i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.270i)T \) |
| 5 | \( 1 + (4.47 + 2.22i)T \) |
| good | 2 | \( 1 + (-0.250 - 0.127i)T + (2.35 + 3.23i)T^{2} \) |
| 7 | \( 1 + (6.72 - 6.72i)T - 49iT^{2} \) |
| 11 | \( 1 + (5.01 + 15.4i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-18.1 + 9.23i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (9.63 - 1.52i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-0.285 + 0.392i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (2.35 - 4.62i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (30.1 + 41.4i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (13.0 + 9.50i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-16.7 - 32.7i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-13.2 + 40.7i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (40.4 + 40.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (0.375 - 2.37i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-24.1 - 3.83i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-20.7 - 6.73i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (19.9 + 61.5i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (57.6 - 9.13i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (27.9 - 20.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-34.0 + 66.7i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-64.6 - 88.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-10.8 - 68.7i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-34.3 + 11.1i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-17.9 + 113. i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−13.49257925566601523984173300273, −12.97558270720902984256790406237, −11.59438074988787507173866616027, −10.67157916869120064075842506297, −9.188683513748435539057893017530, −8.269241731313874208217333993539, −6.18953653446067258645369604694, −5.48936051685726364665308781737, −3.69451374379846567868602574925, −0.37226068459940554623214769095,
3.57074667823934550409989045846, 4.49791505019042890436119757273, 6.71543135560866172177921079779, 7.55957142801378335688759163419, 9.120980545196277932840571731110, 10.47231713223754097350968720860, 11.49355528546749760603048432855, 12.68884630320765040207594409239, 13.32081740247956430456211702911, 14.74171440359760190107534269493
