Question

Here are some experiments for Q2 with $a=1$ done in sage.

The negative even integers appear to be

EDIT Experimentally the zeros of $\zeta(s;N,1)$.

Strangely to me, the zeros \zeta(s;5,1)$ and $\zeta(s;6,1)$ are those of zeta are (the previous revision incorrectly included wrong zeros toocaused by insufficient precision).

The first zeros with of $\Re(s)>\frac{1}{2}$ are close to a line.

Might be a bug in my code and/or sage, but found a zero with real part close \zeta(s;6,2)$ (if computed correctly) don't appear related to $10^8$.

Here is a plot those of $\zeta(s;5,1)$ zeta.

sage code:

import mpmath
mpmath.mp.pretty=True
mpmath.mp.dps=30
mpmath.mp.dps=100
N=5
a=1
an=[0]*N
an[a-1]=1
def L(x):
    return mpmath.dirichlet(x,an)

def search1():
    cac={}
    rr=list(range(-20,0))
    P=[]
    for k in xrange(1,40):
        rr += [0.1 + mpmath.j*k]
    for x in rr:
        try:
            r=mpmath.findroot(L,[x],maxsteps=1000)

        except:  continue
        print r
        ks=str(r)
        if ks in cac:  continue
        cac[ks]=1
        P += [(RR(r.real),RR(r.imag))]
    return P

P=search1()
pt=points(P)
pt.show()