People on this website in many questions regarding functional equation of the zeta function keep telling that the singularities of the gamma function cancels with the zeros of the sin. What does it mean? How do they cancel? How the product of something that is zero with something that is not defined is a finite non-zero number.
And most importantly what Are the numerical answers at those points.
Euler's Reflection Formula: $$\Gamma(1-s)\Gamma(s)=\frac{\pi}{\sin(\pi s)}$$
$$\Gamma(1-s)\sin(\pi s)=\frac{\pi}{\Gamma(s)}$$
Using the Sine Double Angle Formula: $$\Gamma(1-s)2\sin(\frac{\pi s}{2})\cos(\frac{\pi s}{2})=\frac{\pi}{\Gamma(s)}$$
$$\Gamma(1-s)\sin(\frac{\pi s}{2})=\frac{\pi}{2}\sec(\frac{\pi s}{2})\frac{1}{\Gamma(s)}$$
Now you can determine the values as $s\to2n$ since the expression on the righthand side is defined for even integers.