L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (2.25 + 1.30i)5-s − 0.999i·6-s + (−1.49 − 2.18i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.30 + 2.25i)10-s + (3.11 − 1.80i)11-s + (0.499 − 0.866i)12-s + (3.60 − 0.167i)13-s + (−0.199 − 2.63i)14-s − 2.60i·15-s + (−0.5 + 0.866i)16-s + (1.41 + 2.45i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (1.00 + 0.581i)5-s − 0.408i·6-s + (−0.563 − 0.825i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.411 + 0.712i)10-s + (0.940 − 0.542i)11-s + (0.144 − 0.249i)12-s + (0.998 − 0.0463i)13-s + (−0.0532 − 0.705i)14-s − 0.671i·15-s + (−0.125 + 0.216i)16-s + (0.343 + 0.595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19759 + 0.203938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19759 + 0.203938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.49 + 2.18i)T \) |
| 13 | \( 1 + (-3.60 + 0.167i)T \) |
good | 5 | \( 1 + (-2.25 - 1.30i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.11 + 1.80i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.41 - 2.45i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.15 + 0.667i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.46 - 2.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 + (3.46 - 2i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.144 + 0.0835i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.03iT - 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 + (8.43 + 4.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.0 - 5.78i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.01 - 3.49i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 5.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.76iT - 71T^{2} \) |
| 73 | \( 1 + (9.93 - 5.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.37 - 5.83i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.10iT - 83T^{2} \) |
| 89 | \( 1 + (7.07 + 4.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.139iT - 97T^{2} \) |
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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
−10.81103419190643146681288587947, −10.17734928141658044664030710835, −9.010286248091670953120977951387, −7.951153663868860522922348886397, −6.68840029598419343152334290708, −6.42484441880381956176067853268, −5.59082006330284406029483058000, −4.06472436017001798490991289362, −3.08631934826772949382480239354, −1.46806264907537127982586605289,
1.50879604630169235351395754849, 2.90199809108580993390459485303, 4.17868673237760094496089430503, 5.16157657623936781794385943645, 6.05661378031173370597788496387, 6.58967341330131781383305526860, 8.430591161111653311370253649984, 9.400741364269183137599957199313, 9.735800567573428843027775318801, 10.81097629234816118838958441654