I am not sure lol. I just looked a the plot and guessed where the first zero is. The convergence is not very good but I got something like 14.13.

import matplotlib.pyplot as plt
def f(y):
    s=complex(0.5,y)
    sum=0
    for i in range(1,10000):
        sum=sum+(i/((i+1)**s))-((i-s)/(i**s))
    return abs(sum)**2


x=[y/10 for y in range(0, 300)]
y=[f(y/10) for y in range(0, 300)]

plt.plot(x, y)
x=14
h=10**-6
for i in range(0,1000):
    temp=f(x)
    df=(f(x+h)-temp)/h
    if(df==0):
        break
    x=x-temp/df
    
print(x)