L(s) = 1 | − 2·2-s + 3·4-s − 6·5-s − 4·8-s + 12·10-s + 3·16-s + 16·19-s − 18·20-s − 4·23-s + 21·25-s + 12·29-s − 6·32-s − 32·38-s + 24·40-s − 16·43-s + 8·46-s − 8·47-s + 36·49-s − 42·50-s − 8·53-s − 24·58-s + 11·64-s + 48·71-s − 12·73-s + 48·76-s − 18·80-s + 32·86-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 2.68·5-s − 1.41·8-s + 3.79·10-s + 3/4·16-s + 3.67·19-s − 4.02·20-s − 0.834·23-s + 21/5·25-s + 2.22·29-s − 1.06·32-s − 5.19·38-s + 3.79·40-s − 2.43·43-s + 1.17·46-s − 1.16·47-s + 36/7·49-s − 5.93·50-s − 1.09·53-s − 3.15·58-s + 11/8·64-s + 5.69·71-s − 1.40·73-s + 5.50·76-s − 2.01·80-s + 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8480677431\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8480677431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{6} \) |
good | 7 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( 1 - 20 T^{2} + p T^{4} + 1944 T^{6} + p^{3} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 32 T^{2} + 747 T^{4} - 10688 T^{6} + 747 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 22 T^{2} + 895 T^{4} - 12052 T^{6} + 895 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 8 T + 53 T^{2} - 288 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 + 2 T + 37 T^{2} + 60 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 6 T + 59 T^{2} - 340 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 106 T^{2} + 6495 T^{4} - 243052 T^{6} + 6495 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 144 T^{2} + 10235 T^{4} - 459424 T^{6} + 10235 p^{2} T^{8} - 144 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 112 T^{2} + 7987 T^{4} - 366304 T^{6} + 7987 p^{2} T^{8} - 112 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 8 T + 73 T^{2} + 752 T^{3} + 73 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 + 4 T + 49 T^{2} + 24 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 4 T + 87 T^{2} + 72 T^{3} + 87 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 244 T^{2} + 26347 T^{4} - 1820968 T^{6} + 26347 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 182 T^{2} + 14439 T^{4} - 860564 T^{6} + 14439 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 161 T^{2} + 64 T^{3} + 161 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 24 T + 365 T^{2} - 3536 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 6 T + 71 T^{2} + 52 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 138 T^{2} + 18623 T^{4} - 1324204 T^{6} + 18623 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 - 186 T^{2} + 19655 T^{4} - 1523596 T^{6} + 19655 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 416 T^{2} + 78163 T^{4} - 8732480 T^{6} + 78163 p^{2} T^{8} - 416 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 2 T + 87 T^{2} + 820 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−6.23702677586873992934111308307, −6.23514234043085971187256959208, −6.00658638259091708753139456317, −5.81403786327861424284012717952, −5.37638573831189931998924952720, −5.16860538703249658573720200011, −5.02799892708519540520587223526, −4.89351865727131501774244785814, −4.85220506268491266369014046905, −4.78498066038201084240610559041, −4.12012216804413939478467720260, −3.83524111825846302276724745974, −3.83204474239191399918377400902, −3.80848591670168681926778503212, −3.55684621164323180575192779133, −3.23740396396156240183684737019, −3.05397306429230918862682647609, −2.71890742791305965845705278487, −2.59070308067302866349665216807, −2.43685981076376651903883206501, −1.73242681721583557075699740113, −1.66626809346835669928442269973, −1.04838367354528433070263598597, −0.65039299344575439674815142797, −0.63993832267415856731392077254,
0.63993832267415856731392077254, 0.65039299344575439674815142797, 1.04838367354528433070263598597, 1.66626809346835669928442269973, 1.73242681721583557075699740113, 2.43685981076376651903883206501, 2.59070308067302866349665216807, 2.71890742791305965845705278487, 3.05397306429230918862682647609, 3.23740396396156240183684737019, 3.55684621164323180575192779133, 3.80848591670168681926778503212, 3.83204474239191399918377400902, 3.83524111825846302276724745974, 4.12012216804413939478467720260, 4.78498066038201084240610559041, 4.85220506268491266369014046905, 4.89351865727131501774244785814, 5.02799892708519540520587223526, 5.16860538703249658573720200011, 5.37638573831189931998924952720, 5.81403786327861424284012717952, 6.00658638259091708753139456317, 6.23514234043085971187256959208, 6.23702677586873992934111308307
Plot not available for L-functions of degree greater than 10.