To add a little to the excellent answer above - it is known that for $0<a<1, a \ne 1/2$ the Riemann Hurwitz function has a lot of zeroes in the strip $1 < \sigma < 1+a$, so, in particular, there cannot be a simple product representation for $\Re s >1$ of Euler kind, while for $a=1/2$ there is, of course, a product representation for $\Re s >1$ though it has the extra term $(1-2^{-s})$ so it's not quite an Euler product; as for continuation and functional equation (in terms of the Lerch function at $1-s$ though), Apostol book Introduction to Analytic Number Theory has it done very nicely

To add a little to the excellent answer above - it is known that for $0<a<1, a \ne 1/2$ the Riemann Hurwitz function has a lot of zeroes in the strip $1 < \sigma < 1+a$, so, in particular, there cannot be a simple product representation for $\Re s >1$ of Euler kind, while for $a=1/2$ there is, of course, a product representation for $\Re s >1$ though it has the extra term $(1-2^{-s})$ so it's not quite an Euler product; as for continuation and functional equation (in terms of the Lerch function at $1-s$ though), Apostol's book Introduction to Analytic Number Theory has it done very nicely.