| L(s) = 1 | − 3·7-s + 8-s − 3·11-s − 3·13-s − 3·17-s + 9·19-s − 3·23-s − 27-s + 15·29-s + 12·31-s − 18·37-s + 18·41-s + 21·43-s + 24·47-s + 21·49-s + 12·53-s − 3·56-s + 27·59-s − 45·61-s − 12·67-s + 18·71-s + 15·73-s + 9·77-s − 3·79-s − 3·83-s − 3·88-s − 9·89-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.353·8-s − 0.904·11-s − 0.832·13-s − 0.727·17-s + 2.06·19-s − 0.625·23-s − 0.192·27-s + 2.78·29-s + 2.15·31-s − 2.95·37-s + 2.81·41-s + 3.20·43-s + 3.50·47-s + 3·49-s + 1.64·53-s − 0.400·56-s + 3.51·59-s − 5.76·61-s − 1.46·67-s + 2.13·71-s + 1.75·73-s + 1.02·77-s − 0.337·79-s − 0.329·83-s − 0.319·88-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.481483468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.481483468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 3 | \( 1 + T^{3} + T^{6} \) |
| 5 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 - 9 T + 179 T^{3} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( 1 + 3 T - 12 T^{2} - 19 T^{3} + 171 T^{4} + 18 p T^{5} - 1161 T^{6} + 18 p^{2} T^{7} + 171 p^{2} T^{8} - 19 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 3 T - 24 T^{2} - 31 T^{3} + 531 T^{4} + 30 p T^{5} - 6181 T^{6} + 30 p^{2} T^{7} + 531 p^{2} T^{8} - 31 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 3 T + 18 T^{2} + 102 T^{3} + 477 T^{4} + 1533 T^{5} + 6821 T^{6} + 1533 p T^{7} + 477 p^{2} T^{8} + 102 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 3 T + 54 T^{2} + 126 T^{3} + 1665 T^{4} + 177 p T^{5} + 32581 T^{6} + 177 p^{2} T^{7} + 1665 p^{2} T^{8} + 126 p^{3} T^{9} + 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 3 T + 39 T^{2} + 97 T^{3} + 1434 T^{4} + 4122 T^{5} + 35249 T^{6} + 4122 p T^{7} + 1434 p^{2} T^{8} + 97 p^{3} T^{9} + 39 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 15 T + 99 T^{2} - 387 T^{3} - 162 T^{4} + 17112 T^{5} - 132695 T^{6} + 17112 p T^{7} - 162 p^{2} T^{8} - 387 p^{3} T^{9} + 99 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 12 T + 24 T^{2} - 26 T^{3} + 2196 T^{4} - 7092 T^{5} - 25353 T^{6} - 7092 p T^{7} + 2196 p^{2} T^{8} - 26 p^{3} T^{9} + 24 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 9 T + 99 T^{2} + 487 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 18 T + 90 T^{2} + 517 T^{3} - 6795 T^{4} + 4023 T^{5} + 189053 T^{6} + 4023 p T^{7} - 6795 p^{2} T^{8} + 517 p^{3} T^{9} + 90 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 21 T + 132 T^{2} + 226 T^{3} - 4536 T^{4} - 26217 T^{5} + 468693 T^{6} - 26217 p T^{7} - 4536 p^{2} T^{8} + 226 p^{3} T^{9} + 132 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 24 T + 270 T^{2} - 2061 T^{3} + 14139 T^{4} - 95271 T^{5} + 641665 T^{6} - 95271 p T^{7} + 14139 p^{2} T^{8} - 2061 p^{3} T^{9} + 270 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 12 T + 93 T^{2} - 79 T^{3} + 1605 T^{4} - 47529 T^{5} + 579146 T^{6} - 47529 p T^{7} + 1605 p^{2} T^{8} - 79 p^{3} T^{9} + 93 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 27 T + 324 T^{2} - 1930 T^{3} - 2160 T^{4} + 183357 T^{5} - 2019907 T^{6} + 183357 p T^{7} - 2160 p^{2} T^{8} - 1930 p^{3} T^{9} + 324 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 45 T + 882 T^{2} + 9142 T^{3} + 38421 T^{4} - 237357 T^{5} - 4086267 T^{6} - 237357 p T^{7} + 38421 p^{2} T^{8} + 9142 p^{3} T^{9} + 882 p^{4} T^{10} + 45 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 12 T + 54 T^{2} + 705 T^{3} - 81 T^{4} - 30885 T^{5} + 74537 T^{6} - 30885 p T^{7} - 81 p^{2} T^{8} + 705 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 180 T^{2} - 1039 T^{3} + 765 T^{4} + 95067 T^{5} - 1171171 T^{6} + 95067 p T^{7} + 765 p^{2} T^{8} - 1039 p^{3} T^{9} + 180 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 15 T + 120 T^{2} - 680 T^{3} - 3645 T^{4} + 99495 T^{5} - 953967 T^{6} + 99495 p T^{7} - 3645 p^{2} T^{8} - 680 p^{3} T^{9} + 120 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 3 T - 3 T^{2} + 647 T^{3} - 828 T^{4} - 32526 T^{5} - 79587 T^{6} - 32526 p T^{7} - 828 p^{2} T^{8} + 647 p^{3} T^{9} - 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 3 T - 222 T^{2} - 189 T^{3} + 32865 T^{4} + 7500 T^{5} - 3152909 T^{6} + 7500 p T^{7} + 32865 p^{2} T^{8} - 189 p^{3} T^{9} - 222 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 9 T + 252 T^{2} + 1620 T^{3} + 37710 T^{4} + 196695 T^{5} + 3742363 T^{6} + 196695 p T^{7} + 37710 p^{2} T^{8} + 1620 p^{3} T^{9} + 252 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 24 T + 336 T^{2} + 3038 T^{3} + 33912 T^{4} + 449928 T^{5} + 5494947 T^{6} + 449928 p T^{7} + 33912 p^{2} T^{8} + 3038 p^{3} T^{9} + 336 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.73416648119070447452539245585, −5.61723640639435822383588298476, −5.55895120630565437815067054357, −5.38807292867576968396106507784, −4.98587695659416134640466701268, −4.92232895274743741530142983152, −4.47736126460670508978353838089, −4.47390220807510607011636566771, −4.44795998934861232981555917942, −4.41084535118673877238889793271, −3.96266798603349577457080102312, −3.80231476069122600728715484167, −3.53530980784843203903915413623, −3.31344501039702360923875571529, −3.20653929455301551457809703071, −2.82587386145345647755846696205, −2.72808740786159478078645061087, −2.49273022674091842537828904015, −2.35082057719640196427356398328, −2.14773225257693278840125029835, −2.11261080324893219195223820529, −1.10957861463478056120659007418, −1.03759598966275093788417511417, −0.917200015963794142517730741304, −0.49505754036901851570277907216,
0.49505754036901851570277907216, 0.917200015963794142517730741304, 1.03759598966275093788417511417, 1.10957861463478056120659007418, 2.11261080324893219195223820529, 2.14773225257693278840125029835, 2.35082057719640196427356398328, 2.49273022674091842537828904015, 2.72808740786159478078645061087, 2.82587386145345647755846696205, 3.20653929455301551457809703071, 3.31344501039702360923875571529, 3.53530980784843203903915413623, 3.80231476069122600728715484167, 3.96266798603349577457080102312, 4.41084535118673877238889793271, 4.44795998934861232981555917942, 4.47390220807510607011636566771, 4.47736126460670508978353838089, 4.92232895274743741530142983152, 4.98587695659416134640466701268, 5.38807292867576968396106507784, 5.55895120630565437815067054357, 5.61723640639435822383588298476, 5.73416648119070447452539245585
Plot not available for L-functions of degree greater than 10.
