| 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
−11.99734908153922710146560334354, −11.04266868194550581769916543583, −10.02326522492304700392278203877, −9.174593937363945416141260592435, −7.987322736106389816191644102049, −7.54527489791913129715027588236, −5.53094451932138674014068495547, −4.86251419956174866675152029433, −3.77623828866181999698519936145, −2.18215490729703814486525352663,
0.51073726760407568980256667278, 2.06037740425737360845852961257, 4.21033324681212982299264420159, 4.94034334267348763789215360315, 6.21577537395874035539382642490, 7.26635965707646277974020160342, 8.288051890625052363058547076453, 9.311775305434854521438847636311, 10.52974862947566208078395276572, 10.83784565447305394186610180557
