Is there real or complex analytic function whose positive real zeros are the primes? Related to this question

Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named functions
and elementary operations?

If we drop the analytic constraint, from the linked question, $\sin(\pi x)$ vanishes at the integers
and via sums of squares we get the function
$P(x)=\sin^2{(\pi  x)} + \sin^2{( \pi(\Gamma(x)+1) / x)}$
whose positive real zeros are the primes.

Q2 Can we use counting of
zeros to count primes in an interval?

 A: H. Laurent introduced the function
$$f(z)=\sum_{n=1}^\infty(\sin\pi z)^2\left(\frac{1}{n^2\sin\frac{\pi z}{n}}-\frac{1}{\pi n (n-z)}\right)^2$$
whose real roots are the positive primes and have no negative zeros.
(Interméd. Math. 5 (1898) p. 78; 15 (1908) p. 265 [Question 1263]. With a rectification by P. Fatou (Interméd. Math. 16 (1909) p. 248)
I read this on a French translation of a German Encyclopedia that I have not at this moment to give his complete reference.
There is an amusing history of this Encyclopedia here at the University of Sevilla. The hole Encyclopedia and his French translation was acquired in the first years after the Civil War in Spain by the then only professor of mathematics in Seville, Mr. Patricio Peñalver. His successor was my thesis director, D Antonio de Castro Brzezicki. The Encyclopedia was in a cabinet with doors. One day when D. Antonio wanted to consult it, he noticed that it had diminished considerably.  The maid was earning a bonus by selling the encyclopedia by weight as paper.  Curiously, the paper was very light, the kind that yellows with time.
we should not be too critical of the cleaner, hunger is very bad, and the post-war years were terrible in Spain.
One of the volumes that was saved was the translation of P. Bachman's original by J. Hadamard and E. Maillet on "Propositions transcendentes de la théorie des nombres". Of which I have a photocopy at home.
