2 Fixed wrong zeros caused by insufficient precision

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()

1

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

The negative even integers appear to be zeros of $\zeta(s;N,1)$.

Strangely to me, the zeros of zeta are zeros too.

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

Might be a bug in my code and/or sage, but found a zero with real part close to $10^8$.

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

sage code:

import mpmath
mpmath.mp.pretty=True
mpmath.mp.dps=30
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()