I use Mathematica or Maple to calculate the inverse Laplace transform of this function, and can get a result,

In[1]:= eq1 = 
 InverseLaplaceTransform[ (E^-Sqrt[s]) /s^(3/2), s, t] // Simplify // 
  Normal

Out[1]= -1 + (2 E^(-(1/4)/t) Sqrt[t])/Sqrt[\[Pi]] + Erf[1/(2 Sqrt[t])]

However, when I used the residue theorem to calculate this problem, and found that the integral along the small circle around origin (one part of a closed curve) was divergent,

1/(2 \[Pi]I) \!\(
\*SubsuperscriptBox[\(\[Integral]\), 
SubscriptBox[\(r\), \(\[Epsilon]\)], \(\[Placeholder]\)]\(
\*FractionBox[
SuperscriptBox[\(E\), \(-
\*SqrtBox[\(s\)]\)], 
SuperscriptBox[\(s\), \(3/
      2\)]] \[DifferentialD]s\)\), s = \[Epsilon]E^I\[Theta]

Would you like to explain this problem to me? Thanks.