L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (1.94 + 1.12i)5-s − 0.999i·6-s + (−2.03 + 1.69i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.12 − 1.94i)10-s + (0.804 + 1.39i)11-s + (−0.499 + 0.866i)12-s − 1.35·13-s + (2.60 − 0.446i)14-s + 2.24i·15-s + (−0.5 + 0.866i)16-s + (−1.82 + 1.05i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.868 + 0.501i)5-s − 0.408i·6-s + (−0.769 + 0.638i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.354 − 0.614i)10-s + (0.242 + 0.420i)11-s + (−0.144 + 0.249i)12-s − 0.376·13-s + (0.696 − 0.119i)14-s + 0.579i·15-s + (−0.125 + 0.216i)16-s + (−0.443 + 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.681506 + 0.880748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.681506 + 0.880748i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.03 - 1.69i)T \) |
| 19 | \( 1 + (-4.14 - 1.34i)T \) |
good | 5 | \( 1 + (-1.94 - 1.12i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.804 - 1.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + (1.82 - 1.05i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.97 - 5.15i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.0259iT - 29T^{2} \) |
| 31 | \( 1 + (-0.395 - 0.684i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.65 + 0.954i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 + (-1.73 - 1.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.58 - 5.53i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.87 + 4.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.18 - 4.14i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.69 + 5.01i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (6.27 - 3.62i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.3 - 8.26i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.18iT - 83T^{2} \) |
| 89 | \( 1 + (4.57 - 7.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−10.12673970940441080332814492493, −9.730779958826192105811340237464, −9.191056506755601736985075348708, −8.160132433178546046552491608609, −7.10426835007342236353979042680, −6.20371559945048458838521181265, −5.28954079635404767463015160185, −3.81196962488248712820773193059, −2.82609123464645736032270689271, −1.88890544665354239290161118691,
0.63298220050809249290408629714, 1.99298154719371271258614849468, 3.28030582758909217685110185259, 4.78554418115752351654526110779, 5.91463148142323640802271189369, 6.62673921674378077205694998245, 7.41057418429319012882929317785, 8.390252452688787810508775560050, 9.242553398442855686956408812995, 9.755465161897989092786571528746