| L(s) = 1 | − 4.68e5·25-s − 7.50e4·27-s + 4.94e6·49-s − 1.17e5·64-s − 1.16e8·121-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 + ⋯ |
| L(s) = 1 | − 6·25-s − 0.734·27-s + 6·49-s − 0.0562·64-s − 6·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.01511408627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01511408627\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 75092 T^{3} + p^{21} T^{6} \) |
| 23 | \( ( 1 + p^{7} T^{2} )^{3} \) |
| good | 2 | \( ( 1 - 2019 T^{3} + p^{21} T^{6} )( 1 + 2019 T^{3} + p^{21} T^{6} ) \) |
| 5 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 7 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 11 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 13 | \( ( 1 + 84809369662 T^{3} + p^{21} T^{6} )^{2} \) |
| 17 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 19 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 29 | \( ( 1 - 4529019429491982 T^{3} + p^{21} T^{6} )( 1 + 4529019429491982 T^{3} + p^{21} T^{6} ) \) |
| 31 | \( ( 1 - 7613427736873064 T^{3} + p^{21} T^{6} )^{2} \) |
| 37 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 41 | \( ( 1 - 57635558455733574 T^{3} + p^{21} T^{6} )( 1 + 57635558455733574 T^{3} + p^{21} T^{6} ) \) |
| 43 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 47 | \( ( 1 - 359523103737468336 T^{3} + p^{21} T^{6} )( 1 + 359523103737468336 T^{3} + p^{21} T^{6} ) \) |
| 53 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 59 | \( ( 1 - 24708 T + p^{7} T^{2} )^{3}( 1 + 24708 T + p^{7} T^{2} )^{3} \) |
| 61 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 67 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 71 | \( ( 1 - 14777160989538604776 T^{3} + p^{21} T^{6} )( 1 + 14777160989538604776 T^{3} + p^{21} T^{6} ) \) |
| 73 | \( ( 1 - 19699186128365185562 T^{3} + p^{21} T^{6} )^{2} \) |
| 79 | \( ( 1 - p^{7} T^{2} )^{6} \) |
| 83 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 89 | \( ( 1 + p^{7} T^{2} )^{6} \) |
| 97 | \( ( 1 - p^{7} T^{2} )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.72201351025515948888498709232, −6.69588572555970461069963158561, −6.27981283412094986240625700330, −5.97102754789641152637948308714, −5.83654372548428280148044240760, −5.63486617284321988632240351081, −5.52460753319555626815900683318, −5.43933505534388875115507208415, −4.94850909938006679709741140218, −4.45694803116396329460303177334, −4.38733316780046842222768161479, −3.91688134892640730184524312859, −3.86839525236050465292411409734, −3.75387428883566289332722615820, −3.64464443365419623857976543273, −2.94238268434458698608580510949, −2.50750132032375092096440564448, −2.48511157158735619661437952008, −2.03966920362277573817680264111, −2.03921351651026882983297755315, −1.40456869941852292731775402410, −1.38183082205230075816305642799, −0.71647984118915061397246067625, −0.49669506052699704284001331789, −0.01615538672089723281827778351,
0.01615538672089723281827778351, 0.49669506052699704284001331789, 0.71647984118915061397246067625, 1.38183082205230075816305642799, 1.40456869941852292731775402410, 2.03921351651026882983297755315, 2.03966920362277573817680264111, 2.48511157158735619661437952008, 2.50750132032375092096440564448, 2.94238268434458698608580510949, 3.64464443365419623857976543273, 3.75387428883566289332722615820, 3.86839525236050465292411409734, 3.91688134892640730184524312859, 4.38733316780046842222768161479, 4.45694803116396329460303177334, 4.94850909938006679709741140218, 5.43933505534388875115507208415, 5.52460753319555626815900683318, 5.63486617284321988632240351081, 5.83654372548428280148044240760, 5.97102754789641152637948308714, 6.27981283412094986240625700330, 6.69588572555970461069963158561, 6.72201351025515948888498709232
Plot not available for L-functions of degree greater than 10.
