| 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
| \(L(1)\) |
\(\approx\) |
\(5.291505470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.291505470\) |
| \(L(\frac{3}{2})\) |
|
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
−10.64618749814503829067870753305, −9.456118586582519097465682639865, −8.399289248501288264175471708756, −7.26775776102463387477267453413, −6.29339489313863039646455919396, −5.61819967567819413876380319343, −5.04751906510849681136087079073, −3.83789858748315098091179280535, −2.67682171808030688619501530470, −2.13695401119681074116262747267,
2.13695401119681074116262747267, 2.67682171808030688619501530470, 3.83789858748315098091179280535, 5.04751906510849681136087079073, 5.61819967567819413876380319343, 6.29339489313863039646455919396, 7.26775776102463387477267453413, 8.399289248501288264175471708756, 9.456118586582519097465682639865, 10.64618749814503829067870753305
