Actually, there is an even stronger result, often called the interpolation theorem, which follows from a well known theorem of Mittag-Leffler :

Let (z_n) be sequence of complex numbers with no accumulation point. For each n, let l(n) be any integer greater or equal to 1 and (a_nk) (0 <= k <= l(n) ) complex numbers. Then there exists an entire function g(z) such that

g^(k)(z_n) = a_nk

for every n>=1 and every 0 <= k <= l(n)

That is, you can fix values for the derivative at the z_j's.