This is my very first question on this website. I was trying to calculate the value of ζ(4). Although I know the exact value (which I found on google to be pi^4/90) but I wanted to derive it by myself. While doing so, I arrived at this rather peculiar expression: $C = \frac{7ℼ^4}{720} - \frac{1}{2} - \frac{P}{2}$
where C is the value of the composite zeta function at 2 and P is the prime zeta function at 2. My question is this. What will be the value of C? I am getting 0.22090911477. Is this correct?
Assuming
$$\zeta(s)=\sum_{n}\frac{1}{n^s}$$
$$P(s)=\sum_{p\text{ prime}}\frac{1}{p^s}$$
$$\mathcal C(s)=\sum_{n\text{ composite}}\frac{1}{n^s}$$
then
$$\mathcal C(s)=\zeta(s)-P(s)-1$$
and
$$\mathcal C(2)=\zeta(2)-P(2)-1=\frac{\pi^2}{6}-P(2)-1\approx 0.192687$$