https://functions.wolfram.com/07.34.26.0005.01
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Input Form
​​​​​​MeijerG[{{Subscript[a,1],…,Subscript[a,n]},{Subscript[a,n+1],…,Subscript[a,p]}},{{Subscript[b,1],…,Subscript[b,m]},{Subscript[b,m+1],…,Subscript[b,q]}},z]==Sum[((Product[If[j==k,1,Gamma[Subscript[a,k]-Subscript[a,j]]],{j,1,n}]Product[Gamma[1+Subscript[b,j]-Subscript[a,k]],{j,1,m}])/(Product[Gamma[Subscript[a,k]-Subscript[b,j]],{j,m+1,q}]Product[Gamma[1+Subscript[a,j]-Subscript[a,k]],{j,n+1,p}]))z^(Subscript[a,k]-1)HypergeometricPFQ[{1+Subscript[b,1]-Subscript[a,k],…,1+Subscript[b,q]-Subscript[a,k]},{1+Subscript[a,1]-Subscript[a,k],…,1+Subscript[a,k-1]-Subscript[a,k],1+Subscript[a,k+1]-Subscript[a,k],…,1+Subscript[a,p]-Subscript[a,k]},(-1)^(q-m-n)/z],{k,1,n}]/;(p>q||(p==q&&m+n==p+1&&!IntervalMemberQ[Interval[{-1,0}],z])||(p==q&&m+n>p+1)||(p==q&&m+n==p&&Abs[z]>1))&&ForAll[{j,k},Element[{j,k},Integers]&&j!=k&&1<=j<=n&&1<=k<=n,!Element[Subscript[a,j]-Subscript[a,k],Integers]]

Here is where I try to apply it

In[]:=
N[MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-Iπ]]
Out[]=
6.12914×
-8
10
+7.62944×
-8
10

In[]:=
meijerG[a_,b_,z_]=Sum[((Product[If[j==k,1,Gamma[Subscript[a,k]-Subscript[a,j]]],{j,1,n}]Product[Gamma[1+Subscript[b,j]-Subscript[a,k]],{j,1,m}])/(Product[Gamma[Subscript[a,k]-Subscript[b,j]],{j,m+1,q}]Product[Gamma[1+Subscript[a,j]-Subscript[a,k]],{j,n+1,p}]))z^(Subscript[a,k]-1)HypergeometricPFQ[{1+Subscript[b,1]-Subscript[a,k],…,1+Subscript[b,q]-Subscript[a,k]},{1+Subscript[a,1]-Subscript[a,k],…,1+Subscript[a,k-1]-Subscript[a,k],1+Subscript[a,k+1]-Subscript[a,k],…,1+Subscript[a,p]-Subscript[a,k]},(-1)^(q-m-n)/z],{k,1,n}]/;(p>q||(p==q&&m+n==p+1&&!IntervalMemberQ[Interval[{-1,0}],z])||(p==q&&m+n>p+1)||(p==q&&m+n==p&&Abs[z]>1))&&ForAll[{j,k},Element[{j,k},Integers]&&j!=k&&1<=j<=n&&1<=k<=n,!Element[Subscript[a,j]-Subscript[a,k],Integers]]
Out[]=
n
∑
k=1
-1+
a
k
z
HypergeometricPFQ{1-
a
k
+
b
1
,1-
a
k
+
b
q
},{…,1+
a
1
-
a
k
,1+
a
-1+k
-
a
k
,1-
a
k
+
a
1+k
,1-
a
k
+
a
p
},
-m-n+q
(-1)
z

m
∏
j=1
Gamma1-
a
k
+
b
j

n
∏
j=1
Ifjk,1,Gamma[
a
k
-
a
j
]
p
∏
j=1+n
Gamma[1+
a
j
-
a
k
]
q
∏
j=1+m
Gamma
a
k
-
b
j
/;(p>q||(pq&&m+np+1&&!IntervalMemberQ[Interval[{-1,0}],z])||(pq&&m+n>p+1)||(pq&&m+np&&Abs[z]>1))&&
∀
{j,k},{j,k}∈&&j≠k&&1≤j≤n&&1≤k≤n
!
a
j
-
a
k
∈

I get the following.

In[]:=
N[meijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-Iπ]]
NSum
:Limit of summation n is not a number.
NSum
:Limit of summation n is not a number.
NSum
:Limit of summation n is not a number.
General
:Further output of NSum::nslim will be suppressed during this calculation.
Out[]=
NSum
-1+
{{},{1,1,1,1,1,1}}
k
(-π)
HypergeometricPFQ{1-
{{},{1,1,1,1,1,1}}
k
+
{{-4,0,0,0,0,0,0},{}}
1
,1-
{{},{1,1,1,1,1,1}}
k
+
{{-4,0,0,0,0,0,0},{}}
q
},{…,1+
{{},{1,1,1,1,1,1}}
1
-
{{},{1,1,1,1,1,1}}
k
,1+
{{},{1,1,1,1,1,1}}
-1+k
-
{{},{1,1,1,1,1,1}}
k
,1-
{{},{1,1,1,1,1,1}}
k
+
{{},{1,1,1,1,1,1}}
1+k
,1-
{{},{1,1,1,1,1,1}}
k
+
{{},{1,1,1,1,1,1}}
p
},

-m-n+q
(-1)
π

m
∏
j=1
Gamma1-
{{},{1,1,1,1,1,1}}
k
+
{{-4,0,0,0,0,0,0},{}}
j

n
∏
j=1
Ifjk,1,Gamma
{{},{1,1,1,1,1,1}}
k
-
{{},{1,1,1,1,1,1}}
j

p
∏
j=1+n
Gamma1+
{{},{1,1,1,1,1,1}}
j
-
{{},{1,1,1,1,1,1}}
k

q
∏
j=1+m
Gamma
{{},{1,1,1,1,1,1}}
k
-
{{-4,0,0,0,0,0,0},{}}
j
,{k,1,n}/;(p>q||(pq&&m+np+1.&&!IntervalMemberQ[Interval[{-1.,0.}],(0.-1.)3.14159])||(pq&&m+n>p+1.)||(pq&&m+np&&Abs[(0.-1.)3.14159]>1.))&&
∀
{j,k},{j,k}∈&&j≠k&&1.≤j≤n&&1.≤k≤n
!
{{},{1.,1.,1.,1.,1.,1.}}
j
-1.
{{},{1.,1.,1.,1.,1.,1.}}
k
∈