The non-truncated version of this estimate is due to E. Landau, see page 353 of his book "Handbuch der Lehre von der Verteilung der Primzahlen", Teubner, Berlin, 1909. It states that for $s \neq 1$ for which $\zeta(s) \neq 0$ one has
$$(\star)\, {\sum_{n \le x}}'\frac{\Lambda(n)}{n^s} = \frac{x^{1-s}}{1-s}-\frac{\zeta'}{\zeta}(s)-\sum_{\rho}\frac{x^{\rho-s}}{\rho-s} + \sum_{k=1}^{\infty} \frac{x^{-2k-s}}{2k+s},$$
where the $\prime$ in the sum indicates that the last term is counted with weight $1/2$ if $x$ is a positive integer. Here the sum is over non-trivial zeros of $\zeta$, as usual.

This is also given as Exercise 4 in Chapter 12, section 1 of "Multiplicative Number Theory I" by Montgomery--Vaughan.

One should also say that $(\star)$ holds for $s=1$ as well, if one interprets $x^{1-s}/(1-s) - \zeta'(s)/\zeta(s)$ as its limit at $s=1$, which is $\log x -\gamma$.

A truncated version has been used many times in the literature (e.g. Lemma 6 of the paper "On integers free of large prime factors" by A. Hildebrand and G. Tenenbaum, Trans. Am. Math. Soc. 296, 265-290, 1986) and is known to experts. Apart from the blog post Terry Tao mentioned in the comments, and the proof sketch given in GH from MO's answer, a similar proof can also be found in the appendix of the arXiv preprint 2211.08973. It states that uniformly for $\Re s \ge 0$, $x \ge 4$ and $T \ge 2 +3 |\Im s|$ we have
$${\sum_{n \le x}}'\frac{\Lambda(n)}{n^s} = \frac{x^{1-s}}{1-s}-\frac{\zeta'}{\zeta}(s)-\sum_{|\Im(\rho-s)|\le T}\frac{x^{\rho-s}}{\rho-s} + \sum_{k=1}^{\infty} \frac{x^{-2k-s}}{2k+s} + R(x,T)$$
if $\zeta(s)\neq 0$, where $R$ satisfies the bound
$$ R(x,T) \ll (\log x) (x-1)^{-\Re s}\min\left\{ 1, \frac{x}{T\langle x \rangle}\right\} + \frac{\log^2 (xT)}{T}\left( 2^{\Re s } x^{1-\Re s}+\frac{2^{-\Re s}}{\log x}\right)$$
with an absolute implied constant. For $s=0$ this recovers the usual explicit formula (Theorem 12.5 from Montgomery--Vaughan), and its proof is the same. Taking $T \to \infty$ recovers $(\star)$. Here $\langle x \rangle$ is as in the statement of Theorem 12.5: it's the distance of $x$ to the nearest prime power not equal to $x$.

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