Zeta function with Maxima/Macsyma in imaxima.el: Integration formula of zeta

In the very first page of the original Riemann's paper on zeta function, the integration formula of zeta function is presented. This formula is shown below.

Maxima 5.20.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 19f (19F)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) block(load(("/Users/yasube/Programming/imaxima-imath/imaxima.lisp")), linenum:0)$

(%i1) kill(all);

(%i1) "We start from the definition of Gamma function. Instead of using
the name Gamma, we use Pi(s) since Gamma is already defined as a
Maxima function."$

(%i2) f1:Pi(s)='integrate(exp(-x)*x^s,x,0,inf);

(%i3) f2:subst(s-1,s,f1);

(%i4) assume(n>0);

(%i5) f3:changevar(f2,x-n*t,t,x);

(%i6) f4:f3/n^s;

(%i7) "a[n] is an arbitrary number series."$

(%i8) f5:f4*a[n];

(%i9) f6:sum(f5,n,1,inf);

(%i10) "We want to evaluate the right hand side of the previous
equation. We start by exchaging the integration and summation."$

(%i11) f7:integrate(sum(first(rhs(f4))*a[n],n,1,inf),t,0,inf);

(%i12) "Let's define F(x) as follows:"$

(%i13) f71:F(x)=sum(a[n]*x^n,n,1,inf);

(%i14) "Then, we obtain the mellin transformation."$

(%i15) f72:integrate(t^(s-1)*F(exp(-t)),t,0,inf);

(%i16) f8:lhs(f6)=f7;

(%i17) "The simplest application if to assume that a[n] = 1 for all n.
We obtain the zeta function in the left side. Let's see how we
can evaluate the right hand side further."$

(%i18) f9:subst(1,a[n],f8);