| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 2·9-s − 2·10-s + 2·12-s + 4·13-s − 15-s − 4·16-s − 2·17-s − 4·18-s − 2·20-s − 23-s − 4·25-s + 8·26-s − 5·27-s − 2·30-s − 7·31-s − 8·32-s − 4·34-s − 4·36-s + 3·37-s + 4·39-s − 8·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 2/3·9-s − 0.632·10-s + 0.577·12-s + 1.10·13-s − 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s − 0.447·20-s − 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s − 0.365·30-s − 1.25·31-s − 1.41·32-s − 0.685·34-s − 2/3·36-s + 0.493·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.73211325750284141433186231976, −6.81130354135159597032281026635, −6.10008461377510957946279965770, −5.56513823603813445068893148660, −4.72966200734041708156501051889, −3.84763496937519983566731282131, −3.52931135642754093199166167311, −2.68793343495993615580735780031, −1.77464628596756662135858026846, 0,
1.77464628596756662135858026846, 2.68793343495993615580735780031, 3.52931135642754093199166167311, 3.84763496937519983566731282131, 4.72966200734041708156501051889, 5.56513823603813445068893148660, 6.10008461377510957946279965770, 6.81130354135159597032281026635, 7.73211325750284141433186231976
