| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 4·7-s − 8-s + 9-s − 3·10-s − 11-s + 12-s + 2·13-s + 4·14-s + 3·15-s + 16-s + 3·17-s − 18-s − 4·19-s + 3·20-s − 4·21-s + 22-s − 9·23-s − 24-s + 4·25-s − 2·26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.872·21-s + 0.213·22-s − 1.87·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.322251916863395704210651778530, −7.37910337800141495691771151647, −6.51527563032921974711917108602, −6.10596305669162668532233921538, −5.41816605989247562633769847976, −3.95198040403074279429759996861, −3.21147033983875096123804994408, −2.31199426594761228591490841425, −1.59970924998806604854543188825, 0,
1.59970924998806604854543188825, 2.31199426594761228591490841425, 3.21147033983875096123804994408, 3.95198040403074279429759996861, 5.41816605989247562633769847976, 6.10596305669162668532233921538, 6.51527563032921974711917108602, 7.37910337800141495691771151647, 8.322251916863395704210651778530
