infinite product, entire function

Construct an entire function that has simple zeros on the real axis at the points $\displaystyle \pm n^{\frac{1}{4}}, n \geq 0$, and no other zeros.

The back of the book says that $\displaystyle z \prod_{n=1}^{\infty} (1-\frac{z^2}{\sqrt{n}})e^{\frac{z^2}{\sqrt{n}}+\frac {z^4}{2n}}$ works. However, I do not see how to show that this function is entire and satisfies the other conditions. In this section, we covered the Weierstrass Product Theorem; however, I do not think we need to use that here. I need help with this one. Thanks.