L(s) = 1 | − 1.12·2-s + 1.64·3-s − 0.735·4-s + 3.20·5-s − 1.84·6-s − 0.477·7-s + 3.07·8-s − 0.300·9-s − 3.60·10-s − 2.01·11-s − 1.20·12-s + 4.76·13-s + 0.536·14-s + 5.26·15-s − 1.98·16-s + 5.32·17-s + 0.337·18-s + 3.11·19-s − 2.35·20-s − 0.784·21-s + 2.26·22-s − 8.35·23-s + 5.05·24-s + 5.27·25-s − 5.35·26-s − 5.42·27-s + 0.351·28-s + ⋯ |
L(s) = 1 | − 0.795·2-s + 0.948·3-s − 0.367·4-s + 1.43·5-s − 0.754·6-s − 0.180·7-s + 1.08·8-s − 0.100·9-s − 1.13·10-s − 0.607·11-s − 0.348·12-s + 1.32·13-s + 0.143·14-s + 1.35·15-s − 0.497·16-s + 1.29·17-s + 0.0795·18-s + 0.715·19-s − 0.527·20-s − 0.171·21-s + 0.482·22-s − 1.74·23-s + 1.03·24-s + 1.05·25-s − 1.05·26-s − 1.04·27-s + 0.0663·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.474011238\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.474011238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 - T \) |
good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 + 0.477T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 + 8.35T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 + 0.615T + 31T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 - 9.49T + 41T^{2} \) |
| 43 | \( 1 + 6.62T + 43T^{2} \) |
| 47 | \( 1 - 9.76T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 6.65T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 + 8.75T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21279742051422510141997005073, −9.831936623617115227397751390886, −9.120938409099889642949108439660, −8.252509981266536696933405781957, −7.71996714055407512495463070550, −6.12657836382119095863633503913, −5.43849456537974848942931966441, −3.86540515477316040271816725790, −2.61809558359364754313327321479, −1.35785509045111895414181894750,
1.35785509045111895414181894750, 2.61809558359364754313327321479, 3.86540515477316040271816725790, 5.43849456537974848942931966441, 6.12657836382119095863633503913, 7.71996714055407512495463070550, 8.252509981266536696933405781957, 9.120938409099889642949108439660, 9.831936623617115227397751390886, 10.21279742051422510141997005073