| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 8·10-s + 4·11-s − 13-s − 4·16-s − 2·17-s − 19-s + 8·20-s + 8·22-s − 6·23-s + 11·25-s − 2·26-s − 5·31-s − 8·32-s − 4·34-s − 5·37-s − 2·38-s + 6·41-s + 9·43-s + 8·44-s − 12·46-s + 8·47-s + 22·50-s − 2·52-s − 12·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 2.52·10-s + 1.20·11-s − 0.277·13-s − 16-s − 0.485·17-s − 0.229·19-s + 1.78·20-s + 1.70·22-s − 1.25·23-s + 11/5·25-s − 0.392·26-s − 0.898·31-s − 1.41·32-s − 0.685·34-s − 0.821·37-s − 0.324·38-s + 0.937·41-s + 1.37·43-s + 1.20·44-s − 1.76·46-s + 1.16·47-s + 3.11·50-s − 0.277·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.67935615954663, −12.43778360136748, −12.11809031748521, −11.46034540160667, −10.96579154377162, −10.60625924699628, −10.00493634799746, −9.458199389259187, −9.218764209817343, −8.867046896600163, −8.201333944229821, −7.414128696932960, −6.859725051152198, −6.568900401571969, −6.013705240242652, −5.720148935266329, −5.423030485167696, −4.703291244403664, −4.229322948718096, −3.912781864315168, −3.175906486186136, −2.614702967084431, −2.121601377937897, −1.744611493551611, −1.052127198268889, 0,
1.052127198268889, 1.744611493551611, 2.121601377937897, 2.614702967084431, 3.175906486186136, 3.912781864315168, 4.229322948718096, 4.703291244403664, 5.423030485167696, 5.720148935266329, 6.013705240242652, 6.568900401571969, 6.859725051152198, 7.414128696932960, 8.201333944229821, 8.867046896600163, 9.218764209817343, 9.458199389259187, 10.00493634799746, 10.60625924699628, 10.96579154377162, 11.46034540160667, 12.11809031748521, 12.43778360136748, 12.67935615954663
