L(s) = 1 | + 15.0·2-s + 34.0·3-s − 285.·4-s + 1.87e3·5-s + 512.·6-s − 7.04e3·7-s − 1.20e4·8-s − 1.85e4·9-s + 2.82e4·10-s − 2.47e3·11-s − 9.69e3·12-s + 1.03e5·13-s − 1.06e5·14-s + 6.38e4·15-s − 3.49e4·16-s − 5.29e5·17-s − 2.79e5·18-s − 4.31e5·19-s − 5.35e5·20-s − 2.39e5·21-s − 3.73e4·22-s − 1.00e5·23-s − 4.08e5·24-s + 1.56e6·25-s + 1.55e6·26-s − 1.29e6·27-s + 2.00e6·28-s + ⋯ |
L(s) = 1 | + 0.665·2-s + 0.242·3-s − 0.556·4-s + 1.34·5-s + 0.161·6-s − 1.10·7-s − 1.03·8-s − 0.941·9-s + 0.894·10-s − 0.0510·11-s − 0.134·12-s + 1.00·13-s − 0.737·14-s + 0.325·15-s − 0.133·16-s − 1.53·17-s − 0.626·18-s − 0.760·19-s − 0.747·20-s − 0.268·21-s − 0.0339·22-s − 0.0749·23-s − 0.251·24-s + 0.803·25-s + 0.666·26-s − 0.470·27-s + 0.617·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + 3.41e6T \) |
good | 2 | \( 1 - 15.0T + 512T^{2} \) |
| 3 | \( 1 - 34.0T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.87e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.04e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.47e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.03e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.00e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.09e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.41e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.81e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 5.52e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.53e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.01e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.09e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.07e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.81e8T + 7.60e17T^{2} \) |
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
−13.37738714472998474731786260934, −12.83786822040773987254004134609, −10.96279232640493156797588260158, −9.420103992600823713078162375268, −8.826299670333874658772106265025, −6.35344794109548434155918003221, −5.61806596519482826369821693610, −3.77904499124625568549472568998, −2.35468996523677705203841536537, 0,
2.35468996523677705203841536537, 3.77904499124625568549472568998, 5.61806596519482826369821693610, 6.35344794109548434155918003221, 8.826299670333874658772106265025, 9.420103992600823713078162375268, 10.96279232640493156797588260158, 12.83786822040773987254004134609, 13.37738714472998474731786260934