  In this paper, we study the alternating Euler $T$-sums and $\S$-sums, which
are infinite series involving (alternating) odd harmonic numbers, and have
similar forms and close relations to the Dirichlet beta functions. By using the
method of residue computations, we establish the explicit formulas for the
(alternating) linear and quadratic Euler $T$-sums and $\S$-sums, from which,
the parity theorems of Hoffman's double and triple $t$-values and
Kaneko-Tsumura's double and triple $T$-values are further obtained. As
supplements, we also show that the linear $T$-sums and $\S$-sums are
expressible in terms of colored multiple zeta values. Some interesting
consequences and illustrative examples are presented.
