| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 2·11-s − 4·14-s + 5·16-s − 4·22-s − 10·25-s + 6·28-s + 12·29-s − 6·32-s + 4·37-s − 20·43-s + 6·44-s − 3·49-s + 20·50-s + 24·53-s − 8·56-s − 24·58-s + 7·64-s + 16·67-s − 24·71-s − 8·74-s + 4·77-s + 4·79-s + 40·86-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 0.603·11-s − 1.06·14-s + 5/4·16-s − 0.852·22-s − 2·25-s + 1.13·28-s + 2.22·29-s − 1.06·32-s + 0.657·37-s − 3.04·43-s + 0.904·44-s − 3/7·49-s + 2.82·50-s + 3.29·53-s − 1.06·56-s − 3.15·58-s + 7/8·64-s + 1.95·67-s − 2.84·71-s − 0.929·74-s + 0.455·77-s + 0.450·79-s + 4.31·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67970621378441729070222558677, −7.22256681200124913942068687753, −6.86319691613869780218188863622, −6.37260175111975523580528956634, −6.13488512587456810544110169825, −5.40474200258705758947525517831, −5.12309401721916407282462954040, −4.41604514672277597522072986408, −3.93582844976437782803839905400, −3.42781315245391160473547778990, −2.64565898313481655439998497825, −2.25338768348159295239588485846, −1.51420742212859032442326804401, −1.09696957382779221676747338783, 0,
1.09696957382779221676747338783, 1.51420742212859032442326804401, 2.25338768348159295239588485846, 2.64565898313481655439998497825, 3.42781315245391160473547778990, 3.93582844976437782803839905400, 4.41604514672277597522072986408, 5.12309401721916407282462954040, 5.40474200258705758947525517831, 6.13488512587456810544110169825, 6.37260175111975523580528956634, 6.86319691613869780218188863622, 7.22256681200124913942068687753, 7.67970621378441729070222558677
