L(s) = 1 | + 0.456i·2-s + 1.79·4-s + (−1.5 + 0.866i)5-s + (−2.29 − 1.32i)7-s + 1.73i·8-s + (−0.395 − 0.685i)10-s + (−3 + 1.73i)11-s + (−1 + 3.46i)13-s + (0.604 − 1.04i)14-s + 2.79·16-s − 17-s + (−4.58 − 2.64i)19-s + (−2.68 + 1.55i)20-s + (−0.791 − 1.37i)22-s − 0.582·23-s + ⋯ |
L(s) = 1 | + 0.323i·2-s + 0.895·4-s + (−0.670 + 0.387i)5-s + (−0.866 − 0.499i)7-s + 0.612i·8-s + (−0.125 − 0.216i)10-s + (−0.904 + 0.522i)11-s + (−0.277 + 0.960i)13-s + (0.161 − 0.279i)14-s + 0.697·16-s − 0.242·17-s + (−1.05 − 0.606i)19-s + (−0.600 + 0.346i)20-s + (−0.168 − 0.292i)22-s − 0.121·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.114013 + 0.681374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114013 + 0.681374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.29 + 1.32i)T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 - 0.456iT - 2T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + (4.58 + 2.64i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.582T + 23T^{2} \) |
| 29 | \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.708 - 0.409i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.55iT - 37T^{2} \) |
| 41 | \( 1 + (6.08 + 3.51i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.29 - 3.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.29 + 3.05i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.08 + 10.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 9.57iT - 59T^{2} \) |
| 61 | \( 1 + (6.58 - 11.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.16 + 3.55i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.87 - 5.70i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.29 - 9.16i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 + (0.0825 - 0.0476i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.68104763119975624106694997004, −9.934870385051250067953196254230, −8.825192923574887293935389433681, −7.78884439329828323636685647560, −6.95075411061233516151617760267, −6.73436033304535145985452725206, −5.42878572671530588298447478557, −4.22412996180016135864895004713, −3.14235839686318368354568700506, −2.07322861226253045369013194778,
0.30147129766992763349773226462, 2.27164201374704330691778077303, 3.13224830595946569089129920730, 4.17894174282672868760479083984, 5.66881625047677998311853729916, 6.19457343705380935279880467336, 7.44320185302878018332866625172, 8.049799859398400018766564392834, 8.986997270069385768143460481821, 10.19971726657195593690834685080