= Solution
Use a genus-zero <Polyakov path integral>, target signature $(-,+,\ldots,+)$ and all four momenta incoming. Define the <Mandelstam variables> by $s=-(p_1+p_2)^2$, $t=-(p_1+p_3)^2$, $u=-(p_1+p_4)^2$. The <string mass-shell condition> is $p_i^2=4/\alpha'$ and <momentum> conservation is $\sum_i p_i=0$, so
$$
\boxed{s+t+u=-\frac{16}{\alpha'}.}
$$
The reduced amplitude omits the overall momentum-conservation <Dirac delta distribution> and any conventional overall scattering-matrix phase. Interpret each <tachyon vertex operator> as normal ordered; self-contractions must not be included.
The <free-boson worldsheet propagator> is $\langle X^a(z)X^b(w)\rangle=-(\alpha'/2)\eta^{ab}\log|z-w|^2$. <Wick theorem> then evaluates the normal-ordered exponential correlator as the <Koba-Nielsen factor>
$$
\left\langle\prod_{j=1}^4{:}e^{ip_j\cdot X(z_j)}{:}\right\rangle\propto\prod_{i<j}|z_i-z_j|^{\alpha'p_i\cdot p_j}.
$$
The <worldsheet zero mode> supplies <momentum> conservation. Four vertices give $g_s^4$ and the sphere contributes $g_s^{-2}$ through the <string genus expansion>, leaving $g_s^2$.
The sphere's residual <Möbius transformations> fix three insertion points. In <sphere gauge fixing for four string vertices>, take $(z_1,z_2,z_3,z_4)=(z,0,1,\infty)$. The <worldsheet ghost fields>, equivalently the <Faddeev-Popov determinant> for this residual group, supply $|z_{23}z_{24}z_{34}|^2$. At finite $z_4$, this determinant grows as $|z_4|^4$, while the matter correlator decays as $|z_4|^{-4}$ by <momentum> conservation and the tachyon <mass> shell; their product has a finite limit. Equivalently the weight-$(1,1)$ matter operator at infinity is normalized with $|z_4|^4$, while the ghost pair uses the inverse factor. After stripping the fixed-position normalization, only
$$
A^{(4)}=g_s^2\int_{\mathbb C}d^2z\,|z|^{-4-\alpha's/2}|1-z|^{-4-\alpha't/2}
$$
remains. Here choose the standard complex-coordinate measure $d^2z=i\,dz\wedge d\bar z=2\,dx\,dy$. With $dx\,dy$ instead, the reduced normalization must acquire a factor two to give the same requested amplitude. Absolute vertex/sphere normalization is a convention; this choice fixes it consistently with the displayed $2\pi$ prefactor.
Set $a=-1-\alpha's/4$, $b=-1-\alpha't/4$, $c=-1-\alpha'u/4$. Then $a+b+c=1$ and the integrand is $|z|^{2a-2}|1-z|^{2b-2}$. To evaluate the <complex beta integral> first work where $\operatorname{Re}a,\operatorname{Re}b,\operatorname{Re}c>0$. This ensures convergence near $0$, $1$ and infinity. Set $v=1-a$ and $w=1-b$. <Schwinger parameterization> gives
$$
|z|^{-2v}|1-z|^{-2w}=\frac1{\Gamma(v)\Gamma(w)}\int_0^\infty d\lambda\,d\rho\,\lambda^{v-1}\rho^{w-1}e^{-\lambda|z|^2-\rho|z-1|^2}.
$$
Completing the square, the $dx\,dy$ <Gaussian integral> is $\pi(\lambda+\rho)^{-1}\exp[-\lambda\rho/(\lambda+\rho)]$. Change variables to $q=\lambda+\rho$ and $x=\lambda/q$; their <Jacobian determinant> is $q$. Integrating $q$ gives $\Gamma(v+w-1)[x(1-x)]^{-(v+w-1)}$. The remaining integral is the ordinary <beta function> $B(1-w,1-v)=B(b,a)$. Thus
$$
\int_{\mathbb C}dx\,dy\,|z|^{2a-2}|1-z|^{2b-2}
=\pi\frac{\Gamma(a)\Gamma(b)\Gamma(c)}{\Gamma(1-a)\Gamma(1-b)\Gamma(1-c)}.
$$
Restoring $d^2z=2dx\,dy$ and substituting the <Mandelstam variables> proves the <Virasoro–Shapiro amplitude>:
$$
\boxed{A^{(4)}=2\pi g_s^2\frac{\Gamma(-1-\alpha's/4)\Gamma(-1-\alpha't/4)\Gamma(-1-\alpha'u/4)}{\Gamma(2+\alpha's/4)\Gamma(2+\alpha't/4)\Gamma(2+\alpha'u/4)}.}
$$
For physical scattering the original position integral generally fails to converge. The formula defines the amplitude by <analytic continuation> from the convergence domain, with the desired scattering boundary value at real poles; the convergent integral should not be claimed valid for every physical <momentum>.
The <gamma function> is a <meromorphic function>, with <simple poles> at nonpositive integers and no zeros, while its reciprocal $1/\Gamma$ is an <entire function>. Consequently it is a <meromorphic function> of the independent invariants and a <crossing-symmetric scattering amplitude>: permuting $s,t,u$ leaves it unchanged. Its generic channel poles are
$$
\boxed{s=\frac4{\alpha'}(n-1),\quad n=0,1,2,\ldots,}
$$
and the same tower in $t$ and $u$. They represent the exchanged <bosonic string mass spectrum>: the tachyon, massless states including the graviton, and an infinite sequence of massive closed-string levels. There are no threshold <branch cuts> at this tree order.
A precise check of factorization is the <Virasoro–Shapiro amplitude pole residue>. Near $a=-n$, set $c=1+n-b$ in the nonsingular factor. The <gamma function> recurrence gives
$$
\frac{\Gamma(b)\Gamma(1+n-b)}{\Gamma(1-b)\Gamma(b-n)}=(-1)^n\prod_{j=1}^n(b-j)^2.
$$
Combining this with $\Gamma(a)\sim(-1)^n/[n!(a+n)]$ shows
$$
\operatorname*{Res}_{s=4(n-1)/\alpha'}A^{(4)}=-\frac{8\pi g_s^2}{\alpha'(n!)^2}\prod_{j=1}^n(b-j)^2.
$$
The <residue> is a degree-$2n$ polynomial in $t$, consistent with exchange up to spin $2n$ and <scattering-amplitude factorization>. At exceptional kinematics reciprocal Gamma zeros can remove apparent poles. In particular, when both $a$ and $b$ approach nonpositive integers, the zero from $1/\Gamma(a+b)$ cancels the putative double pole, leaving channel simple-pole terms. Thus one must not count products of numerator poles without using $a+b+c=1$.
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