| L(s) = 1 | − 3·4-s − 4·11-s + 4·16-s + 32·19-s + 2·29-s + 10·41-s + 12·44-s − 5·49-s + 28·59-s − 14·61-s − 9·64-s − 8·71-s − 96·76-s − 12·79-s − 60·89-s − 36·101-s − 20·109-s − 6·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.20·11-s + 16-s + 7.34·19-s + 0.371·29-s + 1.56·41-s + 1.80·44-s − 5/7·49-s + 3.64·59-s − 1.79·61-s − 9/8·64-s − 0.949·71-s − 11.0·76-s − 1.35·79-s − 6.35·89-s − 3.58·101-s − 1.91·109-s − 0.557·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536819825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536819825\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.69455030349641775089774965689, −7.24175662027074163879985833183, −7.12338386369203983760094452322, −7.11118042648535941362385945832, −6.92292644948889585028588805074, −6.14821909496671207394796014582, −6.02317889951852103025632616963, −5.68368825407470785520033138161, −5.59384290486228412745919484861, −5.21228152956657950829616000738, −5.19054047779975664598592125918, −5.03556487538940634643289030534, −4.91424739737909736192938338204, −4.25138102848267378950118637080, −4.03289701584901501325640499204, −3.91994113874144726662661087229, −3.56402857927525764084556600102, −3.05509898669358670064665512742, −2.84034043073368269531378945440, −2.77109530669204589411747202351, −2.71976817588939256949472524394, −1.47004028892139664765682362741, −1.24308645829007819880232275434, −1.20513933024391575085177907668, −0.42353469952578296085656667924,
0.42353469952578296085656667924, 1.20513933024391575085177907668, 1.24308645829007819880232275434, 1.47004028892139664765682362741, 2.71976817588939256949472524394, 2.77109530669204589411747202351, 2.84034043073368269531378945440, 3.05509898669358670064665512742, 3.56402857927525764084556600102, 3.91994113874144726662661087229, 4.03289701584901501325640499204, 4.25138102848267378950118637080, 4.91424739737909736192938338204, 5.03556487538940634643289030534, 5.19054047779975664598592125918, 5.21228152956657950829616000738, 5.59384290486228412745919484861, 5.68368825407470785520033138161, 6.02317889951852103025632616963, 6.14821909496671207394796014582, 6.92292644948889585028588805074, 7.11118042648535941362385945832, 7.12338386369203983760094452322, 7.24175662027074163879985833183, 7.69455030349641775089774965689
