  We introduce a new algorithm, which is faster and requires less computing
resources than the ones previously known, to compute the Euler-Kronecker
constants $\mathfrak{G}_q$ for the prime cyclotomic fields
$\mathbb{Q}(\zeta_q)$, where $q$ is an odd prime and $\zeta_q$ is a primitive
$q$-root of unity. With such a new algorithm we evaluated $\mathfrak{G}_q$ and
$\mathfrak{G}_q^+$, where $\mathfrak{G}_q^+$ is the Euler-Kronecker constant of
the maximal real subfield of $\mathbb{Q}(\zeta_q)$, for some very large primes
$q$ thus obtaining two new negative values of $\mathfrak{G}_q$:
$\mathfrak{G}_{9109334831}= -0.248739\dotsc$ and $\mathfrak{G}_{9854964401}=
-0.096465\dotsc$ We also evaluated $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$ for
every odd prime $q\le 10^6$, thus enlarging the size of the previously known
range for $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$. Our method also reveals that
difference $\mathfrak{G}_q - \mathfrak{G}^+_q$ can be computed in a much
simpler way than both its summands, see Section 3.4. Moreover, as a by-product,
we also computed $M_q=\max_{\chi\ne \chi_0} \vert L^\prime/L(1,\chi) \vert $
for every odd prime $q\le 10^6$, where $L(s,\chi)$ are the Dirichlet
$L$-functions, $\chi$ run over the non trivial Dirichlet characters mod $q$ and
$\chi_0$ is the trivial Dirichlet character mod $q$. As another by-product of
our computations, we will also provide more data on the generalised Euler
constants in arithmetic progressions. The programs used to performed the
computations here described and the numerical results obtained are available at
the following web address:
\url{http://www.math.unipd.it/~languasc/EK-comput.html}.
