We give a list of corrections for the paper.

• ∙ p. 186, line 12: when $\unicode[STIX]{x1D708}=0$ , $r^{\unicode[STIX]{x1D708}}$ should be understood to be $r^{\unicode[STIX]{x1D708}}=1$ (even if $r=0$ ). Similarly, when $\unicode[STIX]{x1D708}=0$ , the following values should be understood to be  $1$ :

  1. ° p. 188, $\unicode[STIX]{x1D706}^{\unicode[STIX]{x1D708}}$ in line 16;

  2. ° p. 188, $r^{\unicode[STIX]{x1D708}}$ in line 2 from the bottom;

  3. ° p. 189, $\unicode[STIX]{x1D706}^{\unicode[STIX]{x1D708}}$ in line 2;

  4. ° p. 190, $(z+q)^{\unicode[STIX]{x1D708}}$ in line 9 from the bottom;

  5. ° p. 190, $z^{\unicode[STIX]{x1D708}}$ and $(-{\textstyle \frac{r}{2\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D70F})^{\unicode[STIX]{x1D708}}$ in line 6 from the bottom;

  6. ° p. 190 $({\textstyle \frac{r}{2\unicode[STIX]{x1D6FC}}})^{\unicode[STIX]{x1D708}}$ in line 3 from the bottom;

  7. ° p. 195, $r^{\unicode[STIX]{x1D708}}$ in line 3.

• ∙ p. 187, line 8: insert “ $a(n,s)$ is holomorphic on the same region.” after “ $s\neq 1$ .”

• ∙ p. 195, line 11: “all $s\in \mathbf{C}$ ” should be “ $\Re s>{\textstyle \frac{1}{2}}$ ”.

• ∙ p. 198, Lemma 10(2)(ii) and p. 199, Proposition 3(1)(ii): “ $f_{\ast }^{1+\unicode[STIX]{x1D702}}$ ” should be “ $f_{\ast }^{1+2\unicode[STIX]{x1D702}}$ ”.

• ∙ p. 199, line 15 from the bottom: add “ $M=|D_{K}|=M_{1}$ , $L=1$ and ” after “In this case,”

• ∙ p. 200, line 3: “ $v^{(s-k+1+\unicode[STIX]{x1D708})/2}$ ” should be “ $v^{(\unicode[STIX]{x1D70E}-k+1+\unicode[STIX]{x1D708})/2}$ ”.

• ∙ p. 200, line 8: delete $\unicode[STIX]{x1D70B}$ from the exponent in the power with base $\text{e}$ .

• ∙ p. 200, line 10: “ $\text{e}^{-\unicode[STIX]{x1D70B}(v/2)}$ ” should be “ $\text{e}^{-v/2}$ ”.

• ∙ p. 200, line 11: “ $K_{1}:=2^{\unicode[STIX]{x1D70E}-(1/2)}|\unicode[STIX]{x1D6E4}((s-k+\unicode[STIX]{x1D708}+1)/2)|^{-1}$ ” should be “ $K_{1}:=\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D70E}/2+(1/4)}2^{\unicode[STIX]{x1D70E}+(1/2)}|\unicode[STIX]{x1D6E4}((s-k+\unicode[STIX]{x1D708}+1)/2)|^{-1}$ ”.

• ∙ p. 200, line 9 from the bottom: “ $N_{1}\mid r$ , $N_{2}\nmid r$ ” should be “ $\text{gcd}(r,N)=N_{1}$ ”.

• ∙ p. 204, line 3 from the bottom, insert the following sentence after the formula of $A(1,\pm 10,s)$ : “Similarly, if $r^{2}-100=5f^{2}$ with some $f\in \mathbf{N}$ , then by Propositions 2, 3(2) and Lemma 3(a)(2-2), one has

  $$\begin{eqnarray}\displaystyle & \displaystyle A(1,r,s)=\sqrt{5}\frac{\unicode[STIX]{x1D6F6}_{5,\unicode[STIX]{x1D712}_{5}}^{s}(f)\unicode[STIX]{x1D701}(s)}{(1+5^{-s})\unicode[STIX]{x1D701}(2s)}F_{r/5,1}^{1}(5^{-s}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle F_{r/5,1}^{1}(5^{-s})=\frac{-\unicode[STIX]{x1D712}_{5}(2r/5)5^{-s}}{1-5^{1-2s}}(1-5^{1-s})(1+5^{1-s}-5^{m+1-(2m+1)s}(1+5^{-s})), & \displaystyle \nonumber\end{eqnarray}$$
  where $m$ is the integer such that $5^{m}$ is the highest power of $5$ dividing  $f/5$ .”