Finding roots of a List [closed]

Question

I'm having some problems to find the root of an equation, but I'm solving this equation numerically with a list of numbers. I've got the following equation:

f(s)=1-a(s)+2irho(s)b(s)

where s is a list of complex numbers in the complex plane.

s=List{Re s,Im s}

a(s),b(s) and c(s) are also lists of complex numbers.

I have to find the values of s for which

f(s[Re s,Im s])==0

I've tried using FindRoot, NSolve... but both of them require to have a defined function and not a list of numbers, that's my case. I've also tried to interpolate the list of numbers and trying again with FindRoot or NSolve but they don't seem to work this way. UPDATE: here's the original code

m = 0.13957;
rho[s_] = 
Piecewise[{{Sqrt[1 - 4 m^2/s], Im[s] >= 0}, {-Sqrt[1 - 4 m^2/s], 
 Im[s] < 0}}];
k[s_] := Sqrt[s/4 - m^2];
step = 2/20;
si = Table[(i) step, {i, 1, 21}];
ai[a0_, a1_] = Table[{si[[i]], {a0 + a1 si[[i]]}}, {i, 1, 21}];
im = Table[0, {i, 1, 21}];
tol = 1;
While[tol > 0.001, 
bi[a0_, a1_] = 
Table[ai[a0, a1][[i, 2]], {i, 1, 21}] + 
1/Pi NIntegrate [
Table[si[[i]] im[[i]]/(s (s - si[[i]])), {i, 1, 21}], {s, 4 m^2,
2}, Method -> PrincipalValue, Exclusions -> Thread[si == s], 
AccuracyGoal -> 8];
a[a0_, a1_, b0_] = bi[a0, a1] + I im;
Ima[a0_, a1_, b0_] = 
Table[rho[si[[i]]] b0 (si[[i]] - 4 m^2)^(a0 + a1 si[[i]])/
Abs[Gamma[
a[a0, a1, b0][[i]] + 3/2]] Exp[-a1 si[[
i]] (1 - Log[a1]) + (si[[i]]/
Pi)  NIntegrate[(im[[
i]] Log[(si[[i]] - 4 m^2)/(s - 4 m^2)] + 
Arg[Gamma[a[0.520, 0.902, 0.520][[i]] + 3/2]])/(s (s - 
si[[i]])), {s, 4 m^2, 2}, Method -> PrincipalValue, 
Exclusions -> Thread[si == s], AccuracyGoal -> 8]], {i, 1, 
21}];
tol = Table[Ima[a0, a1, b0][[i]] - im[[i]], {i, 1, 21}];
im = Ima[a0, a1, b0]];

sp = Table[(i step - I i step), {i, 1, 21}];
ialpha[a0_, a1_] = 
Table[a0 + 
a1 sp[[i]] + (sp[[i]]/Pi) NIntegrate[
  Ima[0.520, 0.902, 0.520][[i]]/(s (s - sp[[i]])), {s, 4 m^2, 
   2}], {i, 1, 21}];
beta[a0_, a1_, b0_] = 
Table[Ima[a0, a1, b0][[i, 1]]/rho[sp[[i]]], {i, 1, 21}];

ti[a0_, a1_, b0_] = 
Table[beta[a0, a1, b0][[i]]/(1 - ialpha[a0, a1][[i]]), {i, 1, 21}];
tii[a0_, a1_, b0_] = 
 Table[beta[a0, a1, b0][[
 i]]/(1 - ialpha[a0, a1][[i]] + 
  2 I rho[si[[i]]] beta[a0, a1, b0][[i]]), {i, 1, 21}];
 f[a0_, a1_, b0_] = 
 Table[(1 - ialpha[a0, a1][[i]] + 
 2 I rho[si[[i]]] beta[a0, a1, b0][[i]]), {i, 1, 21}];
 f[0.520, 0.902, 0.520]

So, now that I posted the code (I didn't post it before because it's a bit tedious), I want to know for which values of sp (which is a complex number)

f[0.520, 0.902, 0.520]==0

If anyone could help me.

Thanks.