With mpmath I check also this zero. I think now that my parenthesis "(with some effort)" contains an error. :-/ What it is clear is that the real zeros are those of the function given. But the behavior of this function for t complex is really not simple. I will try to do an X-ray of this function. Later we will try to post it. If I know how to do it.

Yes, the constant $e$ is only to make $x \log(e/x)$ monotone on $[0,1]$. In fact $H_f = H_g$ if $f$ and $g$ coincide on an interval $[0,\varepsilon]$, therefore there is some liberty in choosing $f$.

Perhaps I must explain that the existence of $A\subset{\bf R}$ with $0<H_f(A)<1$ is well known and due to Dvoretzky: Dvoretzky, A. A note on Hausdorff dimension functions. Proc. Cambridge Philos. Soc. 44, (1948). 13–16.

In fact the two sums are equal $\sum_\rho(\rho^{-1}+(1-\rho)^{-1})= \sum_\rho (\rho^{-1}+\overline{\rho}^{-1})$. But I (or Davenport) am speaking about $\zeta(s)$. The case of $L(\chi,s)$ is a little more complicate. In this case the zeros are symmetric with respect to the line $\sigma=\frac12$ but not respect the real axis. The determination of the constant $B(\chi)$ is recent, due to Vorhauer in 2006. Davenport do not include it. You must see the book by Montgomery and Vaughan, Chapter 10. In the sum you must consider the zeros of $L(s,\chi)$ and those of $L(s,\overline{\chi})$.

In Lagarias formula the sum is over the non trivial zeros repeated according to its multiplicity. In fact the formula is the Mittag-Leffler expansion of the meromorphic function $\frac{\hat\zeta'(s)}{\hat\zeta(s)}$ that has poles just at the non-trivial zeros and at $0$ and $1$