Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be irrational by Roger Apéry in 1979.
Do we even know if $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}$$ is an irrational number or not? Is it true that $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}=\frac{a}{b}\sum_{n\geq 1}\frac{(-1)^{k-1}}{n^{3/2}\binom{2n}{n}}?$$ Not sure if Apéry's proof can be adapted here since I haven't read it.
By no means an answer but an extended comment. I should also preface I am an amateur mathematician with no formal education. Following Wikipedia's entry for the Riemann zeta function we have the following functional equation: $$ \zeta(s)=2^{s}\pi^{s-1}\sin\left(\pi s/2\right)\Gamma(1-s)\zeta(1-s)\label{a}\tag{1} $$ Suppose $s=3/2.$ Via direct substitutions I can rewrite RHS side of $\ref{a}$ explicitly as $$ 2^{3/2}\cdot\pi^{3/2-1}\cdot\sin\left(3\pi/4\right)\cdot\Gamma(1-3/2)\cdot\zeta(1-3/2);\label{b}\tag{2} $$ which after plugging in yields, $$ 2^{3/2}\cdot\pi^{3/2-1}\cdot2^{1/2}\cdot(-1)\cdot2^{1}\cdot\pi^{1/2}\cdot\zeta(-1/2).\label{c}\tag{3} $$ After regrouping and gathering like terms one can reduce $\ref{c}$ to, $$ 2^{3/2-1/2+1}\cdot\pi^{3/2-1+1/2}\cdot\zeta(-1/2);\label{d}\tag{4} $$ which is equal to, $$ -4\pi\zeta(-1/2).\label{f}\tag{5} $$ And so $$ \zeta(3/2)=-4\pi\zeta(-1/2), \label{g}\tag{6} $$ Observe: $$ \left(-\frac{1}{4}\right)\cdot\frac{\zeta(3/2)}{\zeta(-1/2)}=\pi. \label{h}\tag{7} $$ Surely $-1/4$ is rational. If each quantities $\zeta(3/2)$ and $\zeta(-1/2)$ are rational then $\pi$ is rational, a contradiction. Subsequently at least one of $\zeta(3/2)$ or $\zeta(-1/2)$, is irrational.
Also following the geometric definition of $\pi$, namely as the ratio of a circle's circumference to its diameter, one has $$ \left(-\frac{1}{4}\right)\cdot\frac{\zeta(3/2)}{\zeta(-1/2)}=\frac{C}{D}=\frac{C}{2r}. \label{i}\tag{8} $$ A quick cancelation shows that, $$ 12.56637061\ldots=-\frac{\zeta(3/2)}{\zeta(-1/2)}=2\cdot\left(\frac{C}{r}\right). \label{j}\tag{9} $$ It's certainly NOT reasonable to interpret the LHS of $\ref{j}$ as twice the ratio of a circle's circumference to its radius - but evidently it is. For example, suppose a circle has radius 4.8, and subsequently circumference 30.1593... then $$ 2\cdot\frac{30.1593\ldots}{4.8}=12.56637061\ldots $$
We use the fact that $\sin(3\pi/4)=2^{-1/2}$ and $\Gamma(1-3/2)=(-1)\cdot2^{1}\cdot\pi^{1/2}.$