| L(s) = 1 | − 2·3-s − 6·7-s + 2·9-s − 4·13-s + 12·21-s − 4·23-s − 6·27-s + 8·39-s − 32·41-s + 18·49-s − 28·53-s − 12·63-s + 8·69-s − 36·73-s − 24·79-s + 11·81-s + 32·89-s + 24·91-s + 12·97-s + 8·101-s − 52·103-s − 28·107-s − 24·109-s − 60·113-s − 8·117-s + 16·121-s + 64·123-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 2/3·9-s − 1.10·13-s + 2.61·21-s − 0.834·23-s − 1.15·27-s + 1.28·39-s − 4.99·41-s + 18/7·49-s − 3.84·53-s − 1.51·63-s + 0.963·69-s − 4.21·73-s − 2.70·79-s + 11/9·81-s + 3.39·89-s + 2.51·91-s + 1.21·97-s + 0.796·101-s − 5.12·103-s − 2.70·107-s − 2.29·109-s − 5.64·113-s − 0.739·117-s + 1.45·121-s + 5.77·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5038 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 12142 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91254053070562514935503473736, −6.64795859632820579460090235973, −6.42208878114882333965518959716, −6.25655267201982712301319864296, −6.17708619993648500037950859593, −5.78757460471766174981593649916, −5.70165169085963980366183265607, −5.57767936108639944451595296675, −5.10540898097233351690398310490, −4.99952184686913169152352280075, −4.91318966044700894105267236931, −4.67200274374457960873776453965, −4.37580328612576462909410838030, −4.02151417175022910978778983921, −3.73494570595563733403600406831, −3.72509765451001374746363636441, −3.41368988455271101343075851536, −3.18289205287173802321553842123, −2.82006796447401445450936316266, −2.66496122735462218499956333935, −2.57943264780482616975443102511, −2.03020804236914376588541850359, −1.47128435472330521694695715814, −1.45136778441717679696762422304, −1.36432430034131910361612659743, 0, 0, 0, 0,
1.36432430034131910361612659743, 1.45136778441717679696762422304, 1.47128435472330521694695715814, 2.03020804236914376588541850359, 2.57943264780482616975443102511, 2.66496122735462218499956333935, 2.82006796447401445450936316266, 3.18289205287173802321553842123, 3.41368988455271101343075851536, 3.72509765451001374746363636441, 3.73494570595563733403600406831, 4.02151417175022910978778983921, 4.37580328612576462909410838030, 4.67200274374457960873776453965, 4.91318966044700894105267236931, 4.99952184686913169152352280075, 5.10540898097233351690398310490, 5.57767936108639944451595296675, 5.70165169085963980366183265607, 5.78757460471766174981593649916, 6.17708619993648500037950859593, 6.25655267201982712301319864296, 6.42208878114882333965518959716, 6.64795859632820579460090235973, 6.91254053070562514935503473736
