I hope this question is well received. I don't have a computer that can calculate very many terms for the infinite series: $$\sum_{n=1}^\infty \frac{(1)^{n+1}}{(2n1)^{2}},$$ but is it going to equal to this closed form: $\log(2.5)$?
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2$\begingroup$ See en.m.wikipedia.org/wiki/Dirichlet_beta_function $\endgroup$ – Sylvain JULIEN May 11 '18 at 20:50
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The sum is equal to Catalan's constant, which is close but not equal to $\log(5/2).$ The difference between the two is $0.000325138$

$\begingroup$ Thank you very much. May your best day of the past be the worst day of your future! Bill $\endgroup$ – user124808 May 11 '18 at 20:26

1$\begingroup$ But, it doesn't appear to have a closed form after looking at the literature, just G? $\endgroup$ – user124808 May 11 '18 at 20:31

2$\begingroup$ I just know that this is a special value of a Dirichlet Lfunction at 2 $\endgroup$ – Bonbon May 11 '18 at 20:42

1$\begingroup$ @user124808 At least it's not known to  the sum you've given is the usual definition of Catalan's constant. But it is an open problem even to determine whether $G$ is a rational number. $\endgroup$ – David Zhang May 11 '18 at 20:47

2$\begingroup$ @GeraldEdgar, since this series is the $L$function of the nontrivial character mod $4$, which is an odd character (the trivial character, which is even, is related to the zetafunction), the value should be thought of as like odd zetavalues rather than even zetavalues, so we shouldn't expect the series to have any simple relation to powers of $\pi$. For comparison, $L(1,\chi_4) = \pi/4$ and $L(3,\chi_4) = \pi^3/32$ are rational multiples of suitable powers of $\pi$. $\endgroup$ – KConrad May 11 '18 at 23:55