L(s) = 1 | − 9·107-s + 9·121-s − 3·125-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 + 233-s + 239-s + 241-s + 251-s + ⋯ |
L(s) = 1 | − 9·107-s + 9·121-s − 3·125-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 + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 419^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 419^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.215731754\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215731754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 419 | \( ( 1 - T )^{9} \) |
good | 3 | \( 1 + T^{9} + T^{18} \) |
| 5 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 7 | \( 1 + T^{9} + T^{18} \) |
| 11 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 13 | \( 1 + T^{9} + T^{18} \) |
| 17 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 19 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 23 | \( 1 + T^{9} + T^{18} \) |
| 29 | \( 1 + T^{9} + T^{18} \) |
| 31 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 37 | \( 1 + T^{9} + T^{18} \) |
| 41 | \( 1 + T^{9} + T^{18} \) |
| 43 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
| 47 | \( 1 + T^{9} + T^{18} \) |
| 53 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 59 | \( 1 + T^{9} + T^{18} \) |
| 61 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 67 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 71 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 73 | \( 1 + T^{9} + T^{18} \) |
| 79 | \( 1 + T^{9} + T^{18} \) |
| 83 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 89 | \( ( 1 - T )^{9}( 1 + T )^{9} \) |
| 97 | \( ( 1 + T^{3} + T^{6} )^{3} \) |
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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
−3.87721824796751087881022593158, −3.67303951443576897549206783202, −3.58527339539763913255401598256, −3.53219083912777896896918187343, −3.37023517317863803666562321540, −3.31574572555050294705032460959, −3.22042173156253992403693613307, −3.19503963367311697645884694499, −2.83255418765032031525044625410, −2.65008713489741765338980161471, −2.60780148605153761526095350510, −2.55764318901993571788248444666, −2.53808315581559930868576070100, −2.47423405537619379461215019819, −2.40652945793503760103474935526, −1.90715463322749938897905488422, −1.88495116373696783686522079848, −1.63562702688464560532315355467, −1.60506382225074005970706581125, −1.53149176527642884379610261094, −1.44195502966376257218302937816, −1.33565728532180028326386106046, −0.72082358672103454948759174130, −0.71646983140580682248183328984, −0.63765598223276998118407857196,
0.63765598223276998118407857196, 0.71646983140580682248183328984, 0.72082358672103454948759174130, 1.33565728532180028326386106046, 1.44195502966376257218302937816, 1.53149176527642884379610261094, 1.60506382225074005970706581125, 1.63562702688464560532315355467, 1.88495116373696783686522079848, 1.90715463322749938897905488422, 2.40652945793503760103474935526, 2.47423405537619379461215019819, 2.53808315581559930868576070100, 2.55764318901993571788248444666, 2.60780148605153761526095350510, 2.65008713489741765338980161471, 2.83255418765032031525044625410, 3.19503963367311697645884694499, 3.22042173156253992403693613307, 3.31574572555050294705032460959, 3.37023517317863803666562321540, 3.53219083912777896896918187343, 3.58527339539763913255401598256, 3.67303951443576897549206783202, 3.87721824796751087881022593158
Plot not available for L-functions of degree greater than 10.