I want to write a Christmas message only with particular zeta values. It is possible?

Question

I want to write a Christmas message to leave as a comment thanking the people who in the next 24th December will solve some of my problems:

I wish you Math Christmas and a Happy New Year ...

Where the dots … will be 2016 written only using particular values of Riemann zeta function.

Example. For the integer $2$, we have $$2=\frac{\zeta^2(2)}{\zeta(4)}-\zeta(0).$$

Question. Can you do the same with the integer $2016$ using sums, addition, subtraction, products, quotients and powers if it is necessary, only with particular values of Riemann zeta function. Thanks in advance.

References:

[1] https://en.wikipedia.org/wiki/Particular_values_of_Riemann_zeta_function

Answer 1

Well we can say that $2016 = 2^{11}-2^{5}$ which means one possibility, of probably many, is
$$2016=\left(\frac{\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)^{11}-\left(\frac{\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)^{5}$$
Also one thing, which I don't know if it fits your criteria, but I think it looks pretty slick, is
$$2016=\sum\limits_{i=1}^{1008}\left(\frac{\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)$$
(It is a pretty point less statement because you could just multiply it by 1008 but hey, it looks cool). Finally
$$2016=\left(\frac{\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)^{5}\left(\frac{\zeta^2(2)}{\zeta(4)}-\zeta(0)\right)^{2}\left(\frac{3\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)$$

Answer 2

What about $$\left(\frac{\zeta^2(2)}{\zeta(4)}+\zeta(0)\right)^{4}-\frac{\zeta^3(-1)}{\zeta^2(-3)\zeta(-7)}$$