| L(s) = 1 | − 4·4-s − 48·7-s + 8·16-s − 32·25-s + 192·28-s + 208·31-s + 932·49-s − 64·64-s − 328·73-s − 192·79-s − 72·97-s + 128·100-s − 16·103-s − 384·112-s − 48·121-s − 832·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 84·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 6.85·7-s + 1/2·16-s − 1.27·25-s + 48/7·28-s + 6.70·31-s + 19.0·49-s − 64-s − 4.49·73-s − 2.43·79-s − 0.742·97-s + 1.27·100-s − 0.155·103-s − 3.42·112-s − 0.396·121-s − 6.70·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.497·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2469179476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2469179476\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T^{2} + p^{3} T^{4} + p^{6} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 16 T^{2} - 58 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 12 T + 127 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 24 T^{2} + 2518 T^{4} + 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 42 T^{2} + 35611 T^{4} + 42 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 920 T^{2} + 370550 T^{4} - 920 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 666 T^{2} + 331099 T^{4} - 666 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 16 p T^{2} + 593510 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 3204 T^{2} + 3979174 T^{4} + 3204 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 52 T + 2346 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 5070 T^{2} + 10169059 T^{4} - 5070 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6644 T^{2} + 16686758 T^{4} - 6644 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4428 T^{2} + 9544198 T^{4} - 4428 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4624 T^{2} + 11292198 T^{4} - 4624 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2868 T^{2} + 15041350 T^{4} + 2868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 9784 T^{2} + 48017174 T^{4} + 9784 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7558 T^{2} + 41065323 T^{4} - 7558 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 638 T^{2} + 32239203 T^{4} + 638 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3316 T^{2} + 50632998 T^{4} - 3316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 82 T + 9539 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 48 T + 12883 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 9988 T^{2} + 90953510 T^{4} + 9988 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1784 T^{2} + 85488374 T^{4} - 1784 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 18 T + 18871 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.31544376198299227527552330736, −5.12646651294273380199777142434, −4.98790262672086296560178454113, −4.94201474282123828977094045817, −4.58734943438911404688538018818, −4.40179883723399128908829166168, −4.33009830852515080798813120186, −4.08398608254383145433613133764, −4.08336660076565381475345405692, −3.81585460312239959793813110181, −3.52990470678851536796206809175, −3.40256915511145817299184563436, −3.38260162893549066161034085553, −2.94655699955912822326257764633, −2.94073067186340627530622753016, −2.81067849036606813900657988214, −2.79173161792555532973673127578, −2.61754347277584465612549158480, −2.47265017319130347722259665676, −1.58867599079603924817395793404, −1.50414731718866756459025161508, −1.22135989553702130014898472700, −0.49987256895242205738017781615, −0.40349510767086770223404778579, −0.22511907150119761399687066159,
0.22511907150119761399687066159, 0.40349510767086770223404778579, 0.49987256895242205738017781615, 1.22135989553702130014898472700, 1.50414731718866756459025161508, 1.58867599079603924817395793404, 2.47265017319130347722259665676, 2.61754347277584465612549158480, 2.79173161792555532973673127578, 2.81067849036606813900657988214, 2.94073067186340627530622753016, 2.94655699955912822326257764633, 3.38260162893549066161034085553, 3.40256915511145817299184563436, 3.52990470678851536796206809175, 3.81585460312239959793813110181, 4.08336660076565381475345405692, 4.08398608254383145433613133764, 4.33009830852515080798813120186, 4.40179883723399128908829166168, 4.58734943438911404688538018818, 4.94201474282123828977094045817, 4.98790262672086296560178454113, 5.12646651294273380199777142434, 5.31544376198299227527552330736
Plot not available for L-functions of degree greater than 10.
