L(s) = 1 | − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s + 45·81-s − 9·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s + 45·81-s − 9·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1759^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1759^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.468914233\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.468914233\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1759 | \( ( 1 - T )^{9} \) |
good | 2 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 3 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 5 | \( 1 + T^{9} + T^{18} \) |
| 7 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 11 | \( 1 + T^{9} + T^{18} \) |
| 13 | \( 1 + T^{9} + T^{18} \) |
| 17 | \( 1 + T^{9} + T^{18} \) |
| 19 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 23 | \( 1 + T^{9} + T^{18} \) |
| 29 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 31 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 37 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 41 | \( 1 + T^{9} + T^{18} \) |
| 43 | \( 1 + T^{9} + T^{18} \) |
| 47 | \( 1 + T^{9} + T^{18} \) |
| 53 | \( 1 + T^{9} + T^{18} \) |
| 59 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 61 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 67 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 71 | \( 1 + T^{9} + T^{18} \) |
| 73 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 79 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 83 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 89 | \( ( 1 + T + T^{2} )^{9} \) |
| 97 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.77928689416932821339181011048, −3.76180282040575080016932319009, −3.74325151887482480483162447904, −3.59263129206123631210820425569, −3.41585794429366520961547592743, −3.38025648270400829781420620684, −3.37649185220190720502312086484, −2.83808812683631469716646747113, −2.83544565113122554110013057887, −2.82567915544817036974319969874, −2.73272695669211770281982213339, −2.42497148044393525087466020339, −2.40266769330274586588337832141, −2.26072148890558489440652546343, −2.15669021924031906706072368420, −2.05217548609731020577389200686, −1.83064701045503901792890330934, −1.79938216648024758676482222416, −1.45591807625374347847185314761, −1.42037068983218769544812341960, −1.19581152659483251335862479881, −1.17620689604478159479229350130, −1.04558241225060459297725624437, −0.968888773396792202436930200982, −0.75367348543036938401868719099,
0.75367348543036938401868719099, 0.968888773396792202436930200982, 1.04558241225060459297725624437, 1.17620689604478159479229350130, 1.19581152659483251335862479881, 1.42037068983218769544812341960, 1.45591807625374347847185314761, 1.79938216648024758676482222416, 1.83064701045503901792890330934, 2.05217548609731020577389200686, 2.15669021924031906706072368420, 2.26072148890558489440652546343, 2.40266769330274586588337832141, 2.42497148044393525087466020339, 2.73272695669211770281982213339, 2.82567915544817036974319969874, 2.83544565113122554110013057887, 2.83808812683631469716646747113, 3.37649185220190720502312086484, 3.38025648270400829781420620684, 3.41585794429366520961547592743, 3.59263129206123631210820425569, 3.74325151887482480483162447904, 3.76180282040575080016932319009, 3.77928689416932821339181011048
Plot not available for L-functions of degree greater than 10.