| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 9-s + 2·10-s − 12·13-s − 16-s + 4·17-s + 18-s − 2·20-s + 3·25-s − 12·26-s − 4·29-s + 5·32-s + 4·34-s − 36-s − 4·37-s − 6·40-s − 12·41-s + 2·45-s + 49-s + 3·50-s + 12·52-s + 20·53-s − 4·58-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 3.32·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.447·20-s + 3/5·25-s − 2.35·26-s − 0.742·29-s + 0.883·32-s + 0.685·34-s − 1/6·36-s − 0.657·37-s − 0.948·40-s − 1.87·41-s + 0.298·45-s + 1/7·49-s + 0.424·50-s + 1.66·52-s + 2.74·53-s − 0.525·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−9.242967464128379637809586382976, −8.447489066361171687346291758504, −7.975429118345566696857391954617, −7.31447554122255167880309018857, −6.97944382340517249810347443111, −6.55815443374401606221787868995, −5.48396360051002103094444371014, −5.37573342552736399524296425305, −5.15942873071656623278244316377, −4.33309058948755389799390879771, −3.89765231085728538463298179402, −2.82363836469554907617267416973, −2.63294756479994443235170715895, −1.63265803663490016612214771155, 0,
1.63265803663490016612214771155, 2.63294756479994443235170715895, 2.82363836469554907617267416973, 3.89765231085728538463298179402, 4.33309058948755389799390879771, 5.15942873071656623278244316377, 5.37573342552736399524296425305, 5.48396360051002103094444371014, 6.55815443374401606221787868995, 6.97944382340517249810347443111, 7.31447554122255167880309018857, 7.975429118345566696857391954617, 8.447489066361171687346291758504, 9.242967464128379637809586382976
