If nobody has a better idea, I will simply get a (real-variable) Taylor series for $\zeta(sigma+it)$ \zeta(\sigma+it)$up to second-order with remainder. This is just (real) calculus - one can easily get the continuous continuation of$\zeta$,$\zeta'$and$\zeta''$up to$Re(s)=1$by Euler-Maclaurin. Perhaps not ideal, but not horrible either. 1 If nobody has a better idea, I will simply get a (real-variable) Taylor series for$\zeta(sigma+it)$up to second-order with remainder. This is just (real) calculus - one can easily get the continuous continuation of$\zeta$,$\zeta'$and$\zeta''$up to$Re(s)=1\$ by Euler-Maclaurin. Perhaps not ideal, but not horrible either.