Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$. Following from , Proposition 2.3 from [Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81(1996), 269-322] and a standard Riemann-Stieltjes partial integration, we plainly have $$\sum_{p\leq x} \frac{|\lambda_f(p)|^2}{p}= \log{\log{x}}+O(1).$$ Is this identity is true: $$\sum_{z<p<w} \frac{|\lambda_f(p)|^2}{p}=\sum_{z<p<w} \frac{1}{p} + o(1),$$ provided that $z,$ $w$ are such that the RHS tends to infinity and $z$ tends to infinity.

Thanks in advance.