  We provide explicit formulas for the integer-valued smooth concordance
invariant $\upsilon(K) = \Upsilon_K(1)$ for every 3-braid knot $K$. We
determine this invariant, which was defined by Ozsv\'ath, Stipsicz and Szab\'o,
by constructing cobordisms between 3-braid knots and (connected sums of) torus
knots. As an application, we show that for positive 3-braid knots $K$ several
alternating distances all equal the sum $g(K) + \upsilon(K)$, where $g(K)$
denotes the 3-genus of $K$. In particular, we compute the alternation number,
the dealternating number and the Turaev genus for all positive 3-braid knots.
We also provide upper and lower bounds on the alternation number and
dealternating number of every 3-braid knot which differ by 1.
