| L(s) = 1 | − 3·5-s − 3·7-s − 6·8-s − 6·11-s − 3·13-s + 6·17-s + 6·23-s + 9·25-s − 3·29-s − 6·31-s + 9·35-s + 30·37-s + 18·40-s + 12·41-s − 12·43-s + 3·49-s + 24·53-s + 18·55-s + 18·56-s + 18·59-s + 3·61-s + 11·64-s + 9·65-s − 6·67-s + 18·73-s + 18·77-s − 6·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 2.12·8-s − 1.80·11-s − 0.832·13-s + 1.45·17-s + 1.25·23-s + 9/5·25-s − 0.557·29-s − 1.07·31-s + 1.52·35-s + 4.93·37-s + 2.84·40-s + 1.87·41-s − 1.82·43-s + 3/7·49-s + 3.29·53-s + 2.42·55-s + 2.40·56-s + 2.34·59-s + 0.384·61-s + 11/8·64-s + 1.11·65-s − 0.733·67-s + 2.10·73-s + 2.05·77-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464505549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464505549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 2 | \( ( 1 + 3 T^{3} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 3 T - 9 T^{3} - 18 T^{4} - 33 T^{5} - 56 T^{6} - 33 p T^{7} - 18 p^{2} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T - 36 T^{3} + 144 T^{4} + 510 T^{5} - 74 T^{6} + 510 p T^{7} + 144 p^{2} T^{8} - 36 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 3 T + 6 T^{2} + 77 T^{3} + 72 T^{4} + 171 T^{5} + 3912 T^{6} + 171 p T^{7} + 72 p^{2} T^{8} + 77 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 3 T + 27 T^{2} - 114 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 6 T - 9 T^{2} + 90 T^{3} - 90 T^{4} + 1146 T^{5} - 8885 T^{6} + 1146 p T^{7} - 90 p^{2} T^{8} + 90 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 3 T - 42 T^{2} - 267 T^{3} + 492 T^{4} + 4359 T^{5} + 9880 T^{6} + 4359 p T^{7} + 492 p^{2} T^{8} - 267 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( ( 1 - 5 T + p T^{2} )^{6} \) |
| 41 | \( 1 - 12 T + 21 T^{2} + 348 T^{3} - 858 T^{4} - 276 p T^{5} + 107233 T^{6} - 276 p^{2} T^{7} - 858 p^{2} T^{8} + 348 p^{3} T^{9} + 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 12 T + 60 T^{2} + 248 T^{3} - 1116 T^{4} - 32580 T^{5} - 266250 T^{6} - 32580 p T^{7} - 1116 p^{2} T^{8} + 248 p^{3} T^{9} + 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 117 T^{2} - 48 T^{3} + 8190 T^{4} + 2808 T^{5} - 435161 T^{6} + 2808 p T^{7} + 8190 p^{2} T^{8} - 48 p^{3} T^{9} - 117 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 12 T + 180 T^{2} - 1266 T^{3} + 180 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 18 T + 135 T^{2} - 258 T^{3} - 3726 T^{4} + 41670 T^{5} - 353441 T^{6} + 41670 p T^{7} - 3726 p^{2} T^{8} - 258 p^{3} T^{9} + 135 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 3 T - 96 T^{2} + 41 T^{3} + 4086 T^{4} + 8577 T^{5} - 220368 T^{6} + 8577 p T^{7} + 4086 p^{2} T^{8} + 41 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T - 96 T^{2} - 292 T^{3} + 5328 T^{4} - 7650 T^{5} - 433722 T^{6} - 7650 p T^{7} + 5328 p^{2} T^{8} - 292 p^{3} T^{9} - 96 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 132 T^{2} - 108 T^{3} + 132 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 9 T + 207 T^{2} - 1298 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 6 T - 132 T^{2} - 364 T^{3} + 10836 T^{4} - 6570 T^{5} - 997698 T^{6} - 6570 p T^{7} + 10836 p^{2} T^{8} - 364 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 12 T - 105 T^{2} - 852 T^{3} + 18294 T^{4} + 780 p T^{5} - 1308125 T^{6} + 780 p^{2} T^{7} + 18294 p^{2} T^{8} - 852 p^{3} T^{9} - 105 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 15 T + 315 T^{2} - 2622 T^{3} + 315 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 12 T - 147 T^{2} - 796 T^{3} + 30726 T^{4} + 612 p T^{5} - 3146871 T^{6} + 612 p^{2} T^{7} + 30726 p^{2} T^{8} - 796 p^{3} T^{9} - 147 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.81539885256370337726395860836, −5.73803842085004445957263297876, −5.61446593163200375632734274681, −5.21185918138214239517269931431, −5.08232964704108697978847940680, −4.85437092119670548201812936995, −4.82692216573693145969741793071, −4.55454872295033852097521776112, −4.34940411409832649015711722055, −4.06905476462822686204282437642, −4.00576633300932013489271437790, −3.69791863539741427957236602525, −3.43009135055179421733649115204, −3.28352927700751383101084401597, −3.24142057702453439162492181676, −2.95722146733724532710690750023, −2.94128592278145188913111063527, −2.44091867320595786766007115056, −2.40776218169233389455132665265, −2.28240781966629604498926252264, −2.07926100239783257622211067417, −1.25837616519868354367468131946, −0.74048924406084730901836778466, −0.66883297642523537982502379660, −0.50382732868667070068770386961,
0.50382732868667070068770386961, 0.66883297642523537982502379660, 0.74048924406084730901836778466, 1.25837616519868354367468131946, 2.07926100239783257622211067417, 2.28240781966629604498926252264, 2.40776218169233389455132665265, 2.44091867320595786766007115056, 2.94128592278145188913111063527, 2.95722146733724532710690750023, 3.24142057702453439162492181676, 3.28352927700751383101084401597, 3.43009135055179421733649115204, 3.69791863539741427957236602525, 4.00576633300932013489271437790, 4.06905476462822686204282437642, 4.34940411409832649015711722055, 4.55454872295033852097521776112, 4.82692216573693145969741793071, 4.85437092119670548201812936995, 5.08232964704108697978847940680, 5.21185918138214239517269931431, 5.61446593163200375632734274681, 5.73803842085004445957263297876, 5.81539885256370337726395860836
Plot not available for L-functions of degree greater than 10.
