You may check two review papers [1]: Dimitrov and Rusev, The zeros of entire Fourier transforms, EAST JOURNAL ON APPROXIMATIONS Volume 17, Number 1 (2011), 1-108 [2]: Ki, The Zeros of Fourier Transforms. For the definition of flatness, you may try $\delta=\int_{-a}^{a}|f'(z)-f'(0)|$. If f(z)=const, then $\delta=0$

@SylvainJULIEN Thanks for comment. I corrected my post to reflect the fact that $\lambda$ is not the De Buijn-Newman constant $\Lambda$. As far as I can remember, Csordas, Odlyzko,Smith,Varga, obtained this lower bound value of $\Lambda$ based on a close pair of numerical zeros of Riemann $\zeta$ function found by Odlyzko.

@Robert: Thanks a lot for you comment. I deleted the parameter $s$ in the questions.

Hutchinson (1923) seemed to be the first one who found such sufficient conditions for polynomials and entire functions to have only real zeros. (J. I. Hutchinson, On a Remarkable Class of Entire Functions , Transactions of the American Mathematical Society, Vol. 25, No. 3 (Jul., 1923), pp.325-332)