In[]:=
ClearAll["`*"];Remove["`*"];
p[x_,t_]:=1-Erf[x/Sqrt[4*t]];pa[x_?NumericQ,t_?NumericQ]:=1
1/4
t
NIntegrate
HypergeometricPFQ{},
1
2
,
3
4
,-
4
s
256t

Gamma
3
4

-
sHypergeometricPFQ{},
3
4
,
5
4
,-
4
s
256t

π
1/4
t
+
2
s
HypergeometricPFQ{},
5
4
,
3
2
,-
4
s
256t

2
t
Gamma
1
4

-
1
4
s
500*Pi*(4^4)t

*p[x,s],{s,0,Infinity},Method->"LocalAdaptive",PrecisionGoal->4(*IncreasethePrecisionGoal.Theresultwillbemoreaccurateattheexpenseofcomputationtime.*)
In[]:=
n=3;ListLinePlot[Table[{t,Log[10,1-pa[1/10,10^t]]},{t,-3,8,n^-1}],AxesLabel->{Log[t],Log[1-P_0.5[x,t]]}](*Increasethenparameter.Thegraphwillbemoreaccurateattheexpenseofcomputationtime.*)
Out[]=
In[]:=
Plot[{Log[10,1-pa[1/10,10^t]]},{t,-3,8},PlotRange->All,AxesLabel->{Log[t],Log[1-P_0.5[x,t]]}](*Veryveryslow!!!*)
Out[]=
$Aborted