The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as:
$$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$
It has the following Euler product (I used that Dirichlet character $\chi_{4}(p)=\sin\left(\frac{p \,\pi}{2}\right)$):
$$\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p \,\pi}{2}\right)} \bigg)$$
Numerical evidence suggests that this Euler product also (slowly) converges for values $\Re(s)>\frac12$.
Does convergence in the domain $\frac12 < \Re(s) \le 1$ indeed occur? If so, could this be proven (I guess a proof would also imply that all complex zeros of $\beta(s)$ must reside on the critical line)?
P.S.:
Just to share that by multiplying the Leibniz formula for $\pi$ ($\beta(1)=\frac{\pi}{4}$) and this formula (35), gives the very elegant relationship:
$$\prod_p \bigg(\frac{p-\sin\left(\frac{p \,\pi}{2}\right)}{{p+\sin\left(\frac{p \,\pi}{2}\right)}} \bigg)=2$$