| L(s) = 1 | + 2-s − 3·3-s + 4-s − 2·5-s − 3·6-s − 2·7-s + 3·9-s − 2·10-s − 3·12-s + 13-s − 2·14-s + 6·15-s + 16-s + 3·18-s + 4·19-s − 2·20-s + 6·21-s − 14·23-s + 7·25-s + 26-s + 2·27-s − 2·28-s − 4·29-s + 6·30-s + 30·31-s + 3·32-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s − 0.755·7-s + 9-s − 0.632·10-s − 0.866·12-s + 0.277·13-s − 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.707·18-s + 0.917·19-s − 0.447·20-s + 1.30·21-s − 2.91·23-s + 7/5·25-s + 0.196·26-s + 0.384·27-s − 0.377·28-s − 0.742·29-s + 1.09·30-s + 5.38·31-s + 0.530·32-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3031892389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3031892389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 19 | \( 1 - 4 T + 17 T^{2} - 136 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - T + T^{3} - p T^{4} - T^{5} + 11 T^{6} - p T^{7} - p^{3} T^{8} + p^{3} T^{9} - p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 34 p T^{7} - 2 p^{2} T^{8} - 2 p^{3} T^{9} - 3 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 + T + 12 T^{2} + 17 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 9 T^{2} - 36 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 223 p T^{7} + 61 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 23 | \( 1 + 14 T + 99 T^{2} + 382 T^{3} + 346 T^{4} - 7726 T^{5} - 57613 T^{6} - 7726 p T^{7} + 346 p^{2} T^{8} + 382 p^{3} T^{9} + 99 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 4 T - 39 T^{2} - 52 T^{3} + 886 T^{4} - 1916 T^{5} - 33907 T^{6} - 1916 p T^{7} + 886 p^{2} T^{8} - 52 p^{3} T^{9} - 39 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 15 T + 144 T^{2} - 29 p T^{3} + 144 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + T + p T^{2} )^{6} \) |
| 41 | \( 1 - 4 T - 75 T^{2} + 100 T^{3} + 3622 T^{4} + 2012 T^{5} - 174751 T^{6} + 2012 p T^{7} + 3622 p^{2} T^{8} + 100 p^{3} T^{9} - 75 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 99 T^{2} + 218 T^{3} + 6207 T^{4} - 6951 T^{5} - 285738 T^{6} - 6951 p T^{7} + 6207 p^{2} T^{8} + 218 p^{3} T^{9} - 99 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 53 | \( 1 - 6 T - 51 T^{2} + 102 T^{3} + 1086 T^{4} + 11334 T^{5} - 107183 T^{6} + 11334 p T^{7} + 1086 p^{2} T^{8} + 102 p^{3} T^{9} - 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 153 T^{2} + 72 T^{3} + 14382 T^{4} - 5508 T^{5} - 969077 T^{6} - 5508 p T^{7} + 14382 p^{2} T^{8} + 72 p^{3} T^{9} - 153 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 34211 p T^{7} + 6133 p^{2} T^{8} - 504 p^{3} T^{9} - 25 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T - 39 T^{2} - 250 T^{3} + 375 T^{4} - 24519 T^{5} - 284658 T^{6} - 24519 p T^{7} + 375 p^{2} T^{8} - 250 p^{3} T^{9} - 39 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 18 T + 99 T^{2} - 234 T^{3} + 306 T^{4} + 55062 T^{5} - 871373 T^{6} + 55062 p T^{7} + 306 p^{2} T^{8} - 234 p^{3} T^{9} + 99 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 91889 p T^{7} + 17041 p^{2} T^{8} + 252 p^{3} T^{9} + 59 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 11 T - 119 T^{2} - 494 T^{3} + 21403 T^{4} + 27611 T^{5} - 1863994 T^{6} + 27611 p T^{7} + 21403 p^{2} T^{8} - 494 p^{3} T^{9} - 119 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 4 T + 169 T^{2} + 472 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 16 T - 87 T^{2} + 424 T^{3} + 39826 T^{4} - 163780 T^{5} - 2350663 T^{6} - 163780 p T^{7} + 39826 p^{2} T^{8} + 424 p^{3} T^{9} - 87 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 14486 p T^{7} + 418 p^{2} T^{8} - 2166 p^{3} T^{9} - 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−8.818723326978585894456721069922, −8.618678728878695840298435504181, −8.178099061397265020894718182557, −8.002195579321495856740009216699, −7.957880378968913562745530172828, −7.77450293761169913236384536454, −7.46105943233515286424869484117, −6.94008622201490174816918985161, −6.90563790575456069500084841087, −6.47121470790871871560557699886, −6.38751472088923433339225279724, −6.23236419385355723520457615136, −6.03785838530267181462580340344, −5.60571460097962418484689675051, −5.57222427665717340383168612318, −5.17781613996977954792779011068, −4.83128594404653082423649151182, −4.51778149694231630428310060811, −4.17700190894317738548401858803, −4.14833953162062799113937923518, −3.72131723958425200607757363456, −3.02711798323333465484932043094, −2.81871623103593173270887540885, −2.70818058035363871303248431601, −1.36896615447309439343496612575,
1.36896615447309439343496612575, 2.70818058035363871303248431601, 2.81871623103593173270887540885, 3.02711798323333465484932043094, 3.72131723958425200607757363456, 4.14833953162062799113937923518, 4.17700190894317738548401858803, 4.51778149694231630428310060811, 4.83128594404653082423649151182, 5.17781613996977954792779011068, 5.57222427665717340383168612318, 5.60571460097962418484689675051, 6.03785838530267181462580340344, 6.23236419385355723520457615136, 6.38751472088923433339225279724, 6.47121470790871871560557699886, 6.90563790575456069500084841087, 6.94008622201490174816918985161, 7.46105943233515286424869484117, 7.77450293761169913236384536454, 7.957880378968913562745530172828, 8.002195579321495856740009216699, 8.178099061397265020894718182557, 8.618678728878695840298435504181, 8.818723326978585894456721069922
Plot not available for L-functions of degree greater than 10.
