L(s) = 1 | − 2.56·2-s − 3-s + 4.57·4-s − 0.592·5-s + 2.56·6-s − 7-s − 6.59·8-s + 9-s + 1.51·10-s + 0.429·11-s − 4.57·12-s − 5.36·13-s + 2.56·14-s + 0.592·15-s + 7.76·16-s − 5.65·17-s − 2.56·18-s − 3.62·19-s − 2.71·20-s + 21-s − 1.10·22-s − 8.63·23-s + 6.59·24-s − 4.64·25-s + 13.7·26-s − 27-s − 4.57·28-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s − 0.265·5-s + 1.04·6-s − 0.377·7-s − 2.33·8-s + 0.333·9-s + 0.480·10-s + 0.129·11-s − 1.32·12-s − 1.48·13-s + 0.685·14-s + 0.153·15-s + 1.94·16-s − 1.37·17-s − 0.604·18-s − 0.831·19-s − 0.606·20-s + 0.218·21-s − 0.234·22-s − 1.80·23-s + 1.34·24-s − 0.929·25-s + 2.69·26-s − 0.192·27-s − 0.864·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01486418929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01486418929\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 0.592T + 5T^{2} \) |
| 11 | \( 1 - 0.429T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 8.63T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 + 0.418T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 + 1.80T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 0.304T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 0.0829T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.631T + 97T^{2} \) |
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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
−8.337383735243725068644784723106, −7.963581109099745075073339726998, −6.96768913335418623681729524819, −6.68869633884675035397381212761, −5.83809607596432033541699768379, −4.73584755029913624786447559451, −3.73833610999066767594978925114, −2.33898964090906744084670830861, −1.84085734991238448435940188005, −0.088177576484602083945831216250,
0.088177576484602083945831216250, 1.84085734991238448435940188005, 2.33898964090906744084670830861, 3.73833610999066767594978925114, 4.73584755029913624786447559451, 5.83809607596432033541699768379, 6.68869633884675035397381212761, 6.96768913335418623681729524819, 7.963581109099745075073339726998, 8.337383735243725068644784723106