| L(s) = 1 | − 6·7-s + 2·13-s − 2·19-s + 6·25-s − 8·31-s − 18·37-s − 16·43-s + 13·49-s − 10·61-s + 22·67-s + 2·73-s − 10·79-s − 12·91-s + 10·97-s + 2·103-s − 28·109-s − 6·121-s + 127-s + 131-s + 12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 0.554·13-s − 0.458·19-s + 6/5·25-s − 1.43·31-s − 2.95·37-s − 2.43·43-s + 13/7·49-s − 1.28·61-s + 2.68·67-s + 0.234·73-s − 1.12·79-s − 1.25·91-s + 1.01·97-s + 0.197·103-s − 2.68·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.05433643313669533911783232246, −9.397687695952564808383582699781, −8.806587663978251937370981617555, −8.655801172150449552881383118942, −7.79497553082532979201942816895, −6.95550481778817521461633350929, −6.65045650092432229207532369718, −6.44248673632066345518308551587, −5.48459428248370426498378218555, −5.10755730919050596352378085939, −3.99250331289836498435902686175, −3.36478379045069074059258347613, −3.09788365205681643264055516753, −1.81832624891237728767312244270, 0,
1.81832624891237728767312244270, 3.09788365205681643264055516753, 3.36478379045069074059258347613, 3.99250331289836498435902686175, 5.10755730919050596352378085939, 5.48459428248370426498378218555, 6.44248673632066345518308551587, 6.65045650092432229207532369718, 6.95550481778817521461633350929, 7.79497553082532979201942816895, 8.655801172150449552881383118942, 8.806587663978251937370981617555, 9.397687695952564808383582699781, 10.05433643313669533911783232246
