| L(s) = 1 | − 2·3-s − 3·4-s + 3·9-s + 6·12-s + 13-s + 5·16-s + 4·17-s − 6·25-s − 4·27-s − 20·29-s − 9·36-s − 2·39-s − 24·43-s − 10·48-s + 2·49-s − 8·51-s − 3·52-s + 12·53-s − 4·61-s − 3·64-s − 12·68-s + 12·75-s + 16·79-s + 5·81-s + 40·87-s + 18·100-s − 36·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 9-s + 1.73·12-s + 0.277·13-s + 5/4·16-s + 0.970·17-s − 6/5·25-s − 0.769·27-s − 3.71·29-s − 3/2·36-s − 0.320·39-s − 3.65·43-s − 1.44·48-s + 2/7·49-s − 1.12·51-s − 0.416·52-s + 1.64·53-s − 0.512·61-s − 3/8·64-s − 1.45·68-s + 1.38·75-s + 1.80·79-s + 5/9·81-s + 4.28·87-s + 9/5·100-s − 3.58·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19773 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19773 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.60769080996183838146178176777, −9.952802346640542280811898231944, −9.621164434933903195530417891960, −9.192740554674130435254875131876, −8.436327130322014357378474405789, −7.86842128247402012013750181867, −7.31644658791899771958167614488, −6.54556545042783689480045128663, −5.65486780279705070512904066158, −5.45644495626393309252824544773, −4.87384726537776797190932459936, −3.80574672742082577656041868750, −3.71205447024015551816570926916, −1.68781683088650348910637576882, 0,
1.68781683088650348910637576882, 3.71205447024015551816570926916, 3.80574672742082577656041868750, 4.87384726537776797190932459936, 5.45644495626393309252824544773, 5.65486780279705070512904066158, 6.54556545042783689480045128663, 7.31644658791899771958167614488, 7.86842128247402012013750181867, 8.436327130322014357378474405789, 9.192740554674130435254875131876, 9.621164434933903195530417891960, 9.952802346640542280811898231944, 10.60769080996183838146178176777
