| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 9-s + 4·14-s + 5·16-s − 2·18-s + 12·23-s + 6·25-s − 6·28-s − 8·29-s − 6·32-s + 3·36-s − 8·37-s − 8·43-s − 24·46-s − 3·49-s − 12·50-s − 4·53-s + 8·56-s + 16·58-s − 2·63-s + 7·64-s − 24·67-s − 12·71-s − 4·72-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1/3·9-s + 1.06·14-s + 5/4·16-s − 0.471·18-s + 2.50·23-s + 6/5·25-s − 1.13·28-s − 1.48·29-s − 1.06·32-s + 1/2·36-s − 1.31·37-s − 1.21·43-s − 3.53·46-s − 3/7·49-s − 1.69·50-s − 0.549·53-s + 1.06·56-s + 2.10·58-s − 0.251·63-s + 7/8·64-s − 2.93·67-s − 1.42·71-s − 0.471·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.511277482343380670875082616936, −7.67080076534670349964439281106, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.65609487669640152451998833504, −6.25650993618347018284029221741, −5.58930669699156824874707798240, −5.00716852164944696644803170219, −4.60406618583419097735195221715, −3.41650850636158117457558100754, −3.34589616522309311397390467846, −2.65274646328445039923916322365, −1.75668697624836907999103677619, −1.13374758634790644463533329227, 0,
1.13374758634790644463533329227, 1.75668697624836907999103677619, 2.65274646328445039923916322365, 3.34589616522309311397390467846, 3.41650850636158117457558100754, 4.60406618583419097735195221715, 5.00716852164944696644803170219, 5.58930669699156824874707798240, 6.25650993618347018284029221741, 6.65609487669640152451998833504, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 7.67080076534670349964439281106, 8.511277482343380670875082616936
