summation – Error handling in infinite beta Dirichlet Bound Convergence

I've tried to calculate the following slowly double convergent sum in Mathematica 11.3

$$ sum _ {k = 1} ^ { infty} left ( frac {1} {(2k-1) (2k + 1)} left ( sum _ {n = 1} ^ { infty} frac {(- 1) ^ {n-1}} {(2 n-1) ^ {2 k-1}} right) right) tag {1} $$

or $ beta (2k-1) = sum _ {n = 1} ^ { infty} frac {(- 1) ^ {n-1}} {(2 n-1) ^ {2 k-1} } $

When using symbolic notation as in (1), Mathematica gives an incorrect symbolic response $ frac {3 zeta (3)} {2 pi ^ 2} $, the correct answer being $ frac {4} { pi ^ 2} frac {7 , zeta (3)} {8} approx0.4262783988 $.

If I use the Sum text function[] instead, it results in numerous recursion and iteration depth errors that have probably been dropped in the symbolic sum.

I contacted Wolfam for clarification on this subject.

The problem I'm facing to avoid a double sum is that the standard output result for the $ beta (k) $ summation in terms of the generalized function Riemann Zeta is not valid for $ k = $ 1. (The well-known result for $ beta (1) $ is $ beta (1) = frac { pi} {4} $)

$$ beta (k) = 2 ^ {2-4k} left ( zeta left (2k-1, frac {1} {4} right) – {zeta left (2k-1 , frac {3} {4} right) right) $$

I do not know how to force the assumptions on Sum[] to change this behavior.

Thoughts?