| L(s) = 1 | + (0.216 + 0.0897i)2-s + (−0.273 − 1.37i)3-s + (−2.78 − 2.78i)4-s + (0.286 + 0.191i)5-s + (0.0641 − 0.322i)6-s + (5.96 − 3.98i)7-s + (−0.713 − 1.72i)8-s + (6.49 − 2.69i)9-s + (0.0449 + 0.0672i)10-s + (−12.8 − 2.56i)11-s + (−3.07 + 4.59i)12-s + (−5.20 + 5.20i)13-s + (1.65 − 0.328i)14-s + (0.184 − 0.446i)15-s + 15.3i·16-s + ⋯ |
| L(s) = 1 | + (0.108 + 0.0448i)2-s + (−0.0911 − 0.458i)3-s + (−0.697 − 0.697i)4-s + (0.0573 + 0.0383i)5-s + (0.0106 − 0.0537i)6-s + (0.852 − 0.569i)7-s + (−0.0891 − 0.215i)8-s + (0.722 − 0.299i)9-s + (0.00449 + 0.00672i)10-s + (−1.17 − 0.233i)11-s + (−0.256 + 0.383i)12-s + (−0.400 + 0.400i)13-s + (0.118 − 0.0234i)14-s + (0.0123 − 0.0297i)15-s + 0.958i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.563i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.825 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.311963 - 1.00999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.311963 - 1.00999i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.216 - 0.0897i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (0.273 + 1.37i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-0.286 - 0.191i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.96 + 3.98i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (12.8 + 2.56i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (5.20 - 5.20i)T - 169iT^{2} \) |
| 19 | \( 1 + (22.2 + 9.23i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-6.25 + 31.4i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (21.4 - 32.1i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (10.5 - 2.09i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (2.37 + 11.9i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-30.0 + 20.1i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-29.2 + 12.1i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (28.8 - 28.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-53.8 - 22.2i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (1.99 + 4.80i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (25.4 + 38.1i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 59.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-7.79 - 39.1i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-9.83 - 6.57i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-114. - 22.7i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-39.1 + 94.6i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-103. - 103. i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-73.5 + 110. i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−10.83784565447305394186610180557, −10.52974862947566208078395276572, −9.311775305434854521438847636311, −8.288051890625052363058547076453, −7.26635965707646277974020160342, −6.21577537395874035539382642490, −4.94034334267348763789215360315, −4.21033324681212982299264420159, −2.06037740425737360845852961257, −0.51073726760407568980256667278,
2.18215490729703814486525352663, 3.77623828866181999698519936145, 4.86251419956174866675152029433, 5.53094451932138674014068495547, 7.54527489791913129715027588236, 7.987322736106389816191644102049, 9.174593937363945416141260592435, 10.02326522492304700392278203877, 11.04266868194550581769916543583, 11.99734908153922710146560334354
