| L(s) = 1 | − 10·4-s − 8·7-s − 62·13-s + 36·16-s + 70·19-s − 226·25-s + 80·28-s − 530·31-s + 268·37-s − 926·43-s − 638·49-s + 620·52-s + 142·61-s + 280·64-s + 1.61e3·67-s − 446·73-s − 700·76-s + 46·79-s + 496·91-s − 2.15e3·97-s + 2.26e3·100-s + 1.87e3·103-s − 3.78e3·109-s − 288·112-s + 1.39e3·121-s + 5.30e3·124-s + 127-s + ⋯ |
| L(s) = 1 | − 5/4·4-s − 0.431·7-s − 1.32·13-s + 9/16·16-s + 0.845·19-s − 1.80·25-s + 0.539·28-s − 3.07·31-s + 1.19·37-s − 3.28·43-s − 1.86·49-s + 1.65·52-s + 0.298·61-s + 0.546·64-s + 2.95·67-s − 0.715·73-s − 1.05·76-s + 0.0655·79-s + 0.571·91-s − 2.25·97-s + 2.25·100-s + 1.78·103-s − 3.32·109-s − 0.242·112-s + 1.04·121-s + 3.83·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 p T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 226 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 31 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9610 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15670 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 29962 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 265 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 134 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6418 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 463 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82754 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 116098 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 125614 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 71 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 809 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 36074 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 223 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1099198 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1402162 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1075 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.34564804891790974709496791063, −11.16334731319888064873733218049, −10.15106203458863590634125553268, −9.822196011098017140901092563316, −9.407591878758206726735184304835, −9.391255733938428520898691051261, −8.326843615446037715297071277596, −8.180418714679743702433966390037, −7.42138731988962521222133897390, −7.03077537735146799875451813822, −6.32614674224839569722344549090, −5.51161299042878092209512857178, −5.17064211892502287879377628551, −4.69282035369179104688385575160, −3.66482577166482915923645990440, −3.62487481502215866839003795049, −2.43237120139552750501555101967, −1.57123755644566585795960106725, 0, 0,
1.57123755644566585795960106725, 2.43237120139552750501555101967, 3.62487481502215866839003795049, 3.66482577166482915923645990440, 4.69282035369179104688385575160, 5.17064211892502287879377628551, 5.51161299042878092209512857178, 6.32614674224839569722344549090, 7.03077537735146799875451813822, 7.42138731988962521222133897390, 8.180418714679743702433966390037, 8.326843615446037715297071277596, 9.391255733938428520898691051261, 9.407591878758206726735184304835, 9.822196011098017140901092563316, 10.15106203458863590634125553268, 11.16334731319888064873733218049, 11.34564804891790974709496791063
