One approach is the brute force method of evaluating at all points at fixed intervals and when it nears zero write value, this can be combined with adaptive step size. Another approach is approximating it with a polynomial of a certain order and using common methods like Newton-Raphson or bisection to find its roots, another way is to approximate it with interpolating polynomials and find its roots instead.
None of the above methods are efficient. How can I find the zeroes of a Bessel function (of 1st kind of a certain order) more efficiently? Are there other methods?