L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 3.18i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (1.59 + 2.75i)10-s + (−3.49 + 2.01i)11-s + (−0.333 − 3.59i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.23 + 2.13i)17-s + (0.595 + 0.343i)19-s + (−2.75 − 1.59i)20-s + (2.01 − 3.49i)22-s + (−2.89 − 5.01i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.42i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.503 + 0.871i)10-s + (−1.05 + 0.608i)11-s + (−0.0925 − 0.995i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.298 + 0.517i)17-s + (0.136 + 0.0788i)19-s + (−0.616 − 0.355i)20-s + (0.429 − 0.744i)22-s + (−0.603 − 1.04i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2850340115\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2850340115\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.333 + 3.59i)T \) |
good | 5 | \( 1 + 3.18iT - 5T^{2} \) |
| 11 | \( 1 + (3.49 - 2.01i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.23 - 2.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.595 - 0.343i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.89 + 5.01i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.940 + 1.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.17iT - 31T^{2} \) |
| 37 | \( 1 + (0.322 - 0.186i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.83 - 3.94i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.48 + 4.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.8iT - 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 + (2.23 + 1.28i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.79 - 11.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.46 + 4.30i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.08 - 5.24i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 16.5iT - 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 9.39iT - 83T^{2} \) |
| 89 | \( 1 + (7.55 - 4.36i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.32 + 3.07i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.664035162161096084024349309923, −8.364523498830015615180403100939, −7.71137382476081835687771232134, −6.67497066283911910327707373661, −5.48766742283374986942222209666, −5.14190007338743622860186631437, −4.17188932054198823413751394452, −2.58172105237319788500376073800, −1.46341374015006688569504604438, −0.12979598714310611133272292368,
1.80345447403369989181308352007, 2.78057870974291459738983821746, 3.52489163144881442656347742041, 4.72294661400769279952543605939, 5.93869452138341778952550578291, 6.73124953328223900912120329247, 7.54950039857254160822179517874, 7.991711883673516266832414454152, 9.176148142653732928285202499686, 9.778939769843414103196373753153