| L(s) = 1 | − 2.60·2-s − 0.439·3-s + 4.77·4-s + 2.58·5-s + 1.14·6-s + 0.0751·7-s − 7.21·8-s − 2.80·9-s − 6.71·10-s + 3.77·11-s − 2.09·12-s + 0.880·13-s − 0.195·14-s − 1.13·15-s + 9.23·16-s + 3.94·17-s + 7.30·18-s + 0.713·19-s + 12.3·20-s − 0.0330·21-s − 9.83·22-s + 1.17·23-s + 3.17·24-s + 1.65·25-s − 2.29·26-s + 2.55·27-s + 0.358·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.253·3-s + 2.38·4-s + 1.15·5-s + 0.466·6-s + 0.0283·7-s − 2.55·8-s − 0.935·9-s − 2.12·10-s + 1.13·11-s − 0.605·12-s + 0.244·13-s − 0.0522·14-s − 0.292·15-s + 2.30·16-s + 0.955·17-s + 1.72·18-s + 0.163·19-s + 2.75·20-s − 0.00720·21-s − 2.09·22-s + 0.245·23-s + 0.647·24-s + 0.331·25-s − 0.449·26-s + 0.490·27-s + 0.0677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7839233898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7839233898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 0.439T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 - 0.0751T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 0.880T + 13T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 - 0.713T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 6.79T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.890675652498915594865213873131, −9.480346757311269639051839366347, −8.644019024279368496052381391847, −7.985267588524141575280291267891, −6.73168927054624673880112868097, −6.25405810389842949871147974718, −5.31770115945511678574165659809, −3.27135060692043018826346560563, −2.05926316384077669739331869299, −0.996219043278179540137700332564,
0.996219043278179540137700332564, 2.05926316384077669739331869299, 3.27135060692043018826346560563, 5.31770115945511678574165659809, 6.25405810389842949871147974718, 6.73168927054624673880112868097, 7.985267588524141575280291267891, 8.644019024279368496052381391847, 9.480346757311269639051839366347, 9.890675652498915594865213873131
