Is there a way to express $ \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}\,\,B_{\frac{1}{2}}\left(n+\frac{1}{2},\frac{1}{2}\right)$

Question

Trying to answer this question, I face the problem of making explicit (even in terms of special functions) $$ \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}\,\,B_{\frac{1}{2}}\left(n+\frac{1}{2},\frac{1}{2}\right)$$ which is $$1.4898960823080587137464656088853266206604466782309\cdots$$

The summation converges very fast

• It is alternating
• The first term is already $1.57080$
• The ratio of two consecutive summands is asymptotic to $\frac 12$

Any idea ?