| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 3·9-s − 2·12-s + 16-s − 2·17-s − 3·18-s + 8·19-s + 2·24-s + 6·25-s − 4·27-s − 32-s + 2·34-s + 3·36-s − 8·38-s − 20·41-s − 8·43-s − 2·48-s − 10·49-s − 6·50-s + 4·51-s + 4·54-s − 16·57-s + 24·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 9-s − 0.577·12-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 1.83·19-s + 0.408·24-s + 6/5·25-s − 0.769·27-s − 0.176·32-s + 0.342·34-s + 1/2·36-s − 1.29·38-s − 3.12·41-s − 1.21·43-s − 0.288·48-s − 1.42·49-s − 0.848·50-s + 0.560·51-s + 0.544·54-s − 2.11·57-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.574236112355339527046262755983, −8.127473857476079203645788388296, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.79169938699478802588819175836, −6.32412925379975931990856588343, −5.63135877622582686929376756064, −5.08606107989467068869473300282, −5.00716852164944696644803170219, −4.11110200931759471218273933763, −3.34589616522309311397390467846, −2.89023851462989053245064085528, −1.75957782335756829082442787579, −1.17874391207982844132640722818, 0,
1.17874391207982844132640722818, 1.75957782335756829082442787579, 2.89023851462989053245064085528, 3.34589616522309311397390467846, 4.11110200931759471218273933763, 5.00716852164944696644803170219, 5.08606107989467068869473300282, 5.63135877622582686929376756064, 6.32412925379975931990856588343, 6.79169938699478802588819175836, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 8.127473857476079203645788388296, 8.574236112355339527046262755983
