| L(s) = 1 | + 2·5-s − 6·13-s − 16·17-s + 11·25-s − 2·29-s + 32·37-s + 10·41-s + 11·49-s − 32·53-s − 14·61-s − 12·65-s − 48·73-s − 32·85-s + 16·89-s + 6·97-s + 26·101-s − 2·113-s + 19·121-s + 38·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.66·13-s − 3.88·17-s + 11/5·25-s − 0.371·29-s + 5.26·37-s + 1.56·41-s + 11/7·49-s − 4.39·53-s − 1.79·61-s − 1.48·65-s − 5.61·73-s − 3.47·85-s + 1.69·89-s + 0.609·97-s + 2.58·101-s − 0.188·113-s + 1.72·121-s + 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.612571858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612571858\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - 19 T^{2} + 240 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 29 T^{2} + 312 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 35 T^{2} + 264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11 T^{2} - 1728 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 53 T^{2} + 600 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 59 T^{2} - 1008 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^3$ | \( 1 - 91 T^{2} + 1392 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.52970715082192336452072885772, −6.29508088475627907043480150340, −6.13135890122518832594659480835, −6.10113660171914714289797788921, −6.03406668867241788693632621365, −5.57482034236152313071981363797, −5.49692634402863454837929506132, −4.87887261889364541594513755129, −4.69760045740092967387864654559, −4.58178110302276955048212056772, −4.54648711055490534349842270889, −4.41597944559203690086540615261, −4.32239428810879200709165230733, −3.84851659548527075717482829865, −3.21515563296148945379289276073, −3.18576335018436617576143580832, −2.78268414960455543237508180676, −2.72399566332890756166767127056, −2.48719024895437509774433618824, −2.08714648210793591305990354467, −2.03880472520197407921652953260, −1.65558859901771308675742444089, −1.15002971321216486637108731758, −0.74691774357681138457567282645, −0.26139803985808761942234670397,
0.26139803985808761942234670397, 0.74691774357681138457567282645, 1.15002971321216486637108731758, 1.65558859901771308675742444089, 2.03880472520197407921652953260, 2.08714648210793591305990354467, 2.48719024895437509774433618824, 2.72399566332890756166767127056, 2.78268414960455543237508180676, 3.18576335018436617576143580832, 3.21515563296148945379289276073, 3.84851659548527075717482829865, 4.32239428810879200709165230733, 4.41597944559203690086540615261, 4.54648711055490534349842270889, 4.58178110302276955048212056772, 4.69760045740092967387864654559, 4.87887261889364541594513755129, 5.49692634402863454837929506132, 5.57482034236152313071981363797, 6.03406668867241788693632621365, 6.10113660171914714289797788921, 6.13135890122518832594659480835, 6.29508088475627907043480150340, 6.52970715082192336452072885772
