Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$ denote the $k$-th derivative of Riemann zeta function. For real $x$, let $[x]$ denote the nearest integer to $x$.

Conjecture 1: For all $n \ge 1,[\zeta^{(n)}(1/2)]= -2^{n+1} n!$

Conjecture 2: For all $k \ge 1,B \ge 2$ we have $[\zeta^{(k)}(1-1/B)]= -B^{k+1}\cdot k!$ and $\zeta^{(k)}(1-1/B)$ is very close to integer.

Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$ denote the $k$-th derivative of Riemann zeta function. For real $x$, let $[x]$ denote the nearest integer to $x$.

Conjecture 1: For all $n \ge 1,[\zeta^{(n)}(1/2)]= -2^{n+1} n!$

Conjecture 2: For all $k \ge 1,B \ge 2$ we have $[\zeta^{(k)}(1-1/B)]= -B^{k+1}\cdot k!$ and $\zeta^{(k)}(1-1/B)$ is very close to integer.

sagemath code per request:

def zetania2(LK=10,prec=100,bb=[3,5]):
    """
    zeta^(k)(1-1/B), near to integers
    """
    import mpmath
    mpmath.mp.pretty=1
    mpmath.mp.dps=prec
    pre2=10
    for B in bb:
        l0=[]
        lerr=[]
        for k in range(1,LK+1):
            T=mpmath.zeta(1-1/mpmath.mpf(B),derivative=k)
            N=mpmath.nint(T)
            N2= -B**(k+1) * factorial(k)
            l0 += [N-N2]
        print(B,l0)

Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$ denote the $k$-th derivative of Riemann zeta function. For real $x$, let $[x]$ denote the nearest integer to $x$.

Conjecture 1: For all $n \ge 1,[\zeta^{(n)}(1/2)]= -2^{n+1} n!$

Conjecture 2: For all $k \ge 1,B \ge 2, [\zeta^{(k)}(1-1/B)]$ is divisible by $B$ and $\zeta^{(k)}(1-1/B)$ is very close to integer.

Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$ denote the $k$-th derivative of Riemann zeta function. For real $x$, let $[x]$ denote the nearest integer to $x$.

Conjecture 1: For all $n \ge 1,[\zeta^{(n)}(1/2)]= -2^{n+1} n!$

Conjecture 2: For all $k \ge 1,B \ge 2$ we have $[\zeta^{(k)}(1-1/B)]= -B^{k+1}\cdot k!$ and $\zeta^{(k)}(1-1/B)$ is very close to integer.