On fractional moments of the mollified Dirichlet $L$-functions

Let $\chi_1(n)$ be the character of Dirichlet mod 5 such that $\chi_1(2)=i$,
$$ \varkappa=\frac{\sqrt{10-2 \sqrt{5}}-2}{\sqrt{5}-1}. $$

Davenport–Heilbronn function is defined below
$$ f(s)=\frac{1-i\varkappa}{2}L(s,\chi_1)+\frac{1+i\varkappa}{2}L(s,\overline{\chi}_1). $$

The function $f(s)$ was introduced and investigated by Davenport and Heilbronn, in 1936. It satisfies the functional equation of Riemann's type
$$ g(s)=g(1-s), $$
where $g(s)=(\frac{\pi}{5})^{-s/2}\Gamma(\frac{1+s}{2})f(s)$.
It is well-known however, that not all non-trivial zeros of $f(s)$ lie on the line $\Re s=\frac{1}{2}$.
In the region $\Re s>1$, $0<\Im s\le T$ the number of zeros of $f(s)$ exceeds $cT$, where $c>0$ is an absolute constant (Davenport and Heilbronn, 1936).
Moreover, the number of zeros of $f(s)$ in the region $\frac{1}{2}<\sigma_1<\Re s<\sigma_2$, $0<\Im s\le T$ exceeds $c_1T$, where $c>0$ is an absolute constant(S. M. Voronin, 1976).
In 1980, S. M. Voronin proved that «abnormally many» zeros of $f(s)$ lied on the critical line $\Re s=\frac{1}{2}$. Let $N_{0,f}(T)$ be the number of zeros of $f(s)$ on the segment $\Re s=\frac{1}{2}$, $0<\Im s\le T$. S. M. Voronin got the estimate
$$ N_{0,f}(T)>c_2T\exp\{\frac{1}{20}\sqrt{\log\log\log\log T}\}, $$
where $c_2>0$ is an absolute constant.
In 1990, A. A. Karatsuba significantly improved Voronin's estimate and got the inequality
$$ N_{0,f}(T)>T(\log T)^{1/2-\varepsilon}, $$
where $\varepsilon>0$ is an arbitrary small constant, $T>T_0(\varepsilon)>0$.
In 1994, A. A. Karatsuba got somewhat more accurate estimate
$$ N_{0,f}(T)>T(\log T)^{1/2}\exp\{-c_3\sqrt{\log\log T}\}, $$
where $c_3>0$ is an absolute constant.
In 2017, the author got the following estimate
$$ N_{0,f}(T)> T (\log T)^{1/2+1/16-\varepsilon}\quad (\varepsilon>0). $$

In this paper we obtain new upper and lower estimates of the fractional moments of mollified Dirichlet series, from which it follows that
$$ N_{0,f}(T)> T (\log T)^{1/2+1/12-\varepsilon}\quad (\varepsilon>0). $$