| L(s) = 1 | + 2·3-s + 3·5-s − 3·7-s + 6·11-s + 3·13-s + 6·15-s + 6·19-s − 6·21-s + 6·23-s + 9·25-s − 5·27-s + 15·29-s + 3·31-s + 12·33-s − 9·35-s − 6·37-s + 6·39-s + 6·41-s − 3·43-s + 15·47-s + 3·49-s − 36·53-s + 18·55-s + 12·57-s + 3·59-s + 6·61-s + 9·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s − 1.13·7-s + 1.80·11-s + 0.832·13-s + 1.54·15-s + 1.37·19-s − 1.30·21-s + 1.25·23-s + 9/5·25-s − 0.962·27-s + 2.78·29-s + 0.538·31-s + 2.08·33-s − 1.52·35-s − 0.986·37-s + 0.960·39-s + 0.937·41-s − 0.457·43-s + 2.18·47-s + 3/7·49-s − 4.94·53-s + 2.42·55-s + 1.58·57-s + 0.390·59-s + 0.768·61-s + 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.390329508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.390329508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 4 T^{2} - p T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 5 | \( 1 - 3 T + 3 p T^{3} - 27 T^{4} + 6 T^{5} + 61 T^{6} + 6 p T^{7} - 27 p^{2} T^{8} + 3 p^{4} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T + 30 T^{3} + 162 T^{4} - 402 T^{5} - 821 T^{6} - 402 p T^{7} + 162 p^{2} T^{8} + 30 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( ( 1 + 18 T^{2} + 9 T^{3} + 18 p T^{4} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 6 T - 18 T^{2} + 30 T^{3} + 612 T^{4} + 2172 T^{5} - 30449 T^{6} + 2172 p T^{7} + 612 p^{2} T^{8} + 30 p^{3} T^{9} - 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 15 T + 90 T^{2} - 411 T^{3} + 2205 T^{4} - 4110 T^{5} - 17723 T^{6} - 4110 p T^{7} + 2205 p^{2} T^{8} - 411 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 3 T - 6 T^{2} + 221 T^{3} - 639 T^{4} - 2088 T^{5} + 61647 T^{6} - 2088 p T^{7} - 639 p^{2} T^{8} + 221 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 6 T - 90 T^{2} + 210 T^{3} + 7812 T^{4} - 8952 T^{5} - 340301 T^{6} - 8952 p T^{7} + 7812 p^{2} T^{8} + 210 p^{3} T^{9} - 90 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T - 12 T^{2} + 11 p T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 4176 p T^{7} + 153 p^{2} T^{8} + 11 p^{4} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 15 T + 48 T^{2} + 3 T^{3} + 3075 T^{4} - 18798 T^{5} + 4399 T^{6} - 18798 p T^{7} + 3075 p^{2} T^{8} + 3 p^{3} T^{9} + 48 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 18 T + 198 T^{2} + 1521 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 3 T - 114 T^{2} + 501 T^{3} + 6567 T^{4} - 20406 T^{5} - 323957 T^{6} - 20406 p T^{7} + 6567 p^{2} T^{8} + 501 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T - 102 T^{2} + 698 T^{3} + 6048 T^{4} - 26604 T^{5} - 259509 T^{6} - 26604 p T^{7} + 6048 p^{2} T^{8} + 698 p^{3} T^{9} - 102 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 16668 p T^{7} + 14148 p^{2} T^{8} + 446 p^{3} T^{9} - 138 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 15 T + 231 T^{2} + 1833 T^{3} + 231 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 14634 p T^{7} + 2439 p^{2} T^{8} + 1787 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 18 T + 198 T^{3} + 23814 T^{4} - 132750 T^{5} - 711245 T^{6} - 132750 p T^{7} + 23814 p^{2} T^{8} + 198 p^{3} T^{9} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 6 T + 72 T^{2} + 21 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 15 T - 84 T^{2} + 1139 T^{3} + 22203 T^{4} - 134028 T^{5} - 1218567 T^{6} - 134028 p T^{7} + 22203 p^{2} T^{8} + 1139 p^{3} T^{9} - 84 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.68582092825434791126346096798, −6.41679944726593406557584588540, −6.15693197229532623232883241238, −6.13654817495053243056019956516, −5.94018485318890720140628828699, −5.91962011969041426407445239974, −5.38033739547577080378556808304, −5.37604648533010097552585100069, −5.04327689127611963480938657351, −4.79467338124890664422881234876, −4.68399406015828843288797991777, −4.42135656551607590398068520042, −4.27846002097281827006952561126, −3.86315390451844595562142974679, −3.51998903137177942048192118988, −3.50510187568248245447930128917, −3.19786250450417210589000744941, −3.03166646473901264076245437642, −2.92378284490732128024108184440, −2.54449787003725554882635162276, −2.34944387046981384583710587024, −1.94399917305330551439172466061, −1.44567093378340039827623271809, −1.08974351314004192448498425590, −1.06761587383282006321810434411,
1.06761587383282006321810434411, 1.08974351314004192448498425590, 1.44567093378340039827623271809, 1.94399917305330551439172466061, 2.34944387046981384583710587024, 2.54449787003725554882635162276, 2.92378284490732128024108184440, 3.03166646473901264076245437642, 3.19786250450417210589000744941, 3.50510187568248245447930128917, 3.51998903137177942048192118988, 3.86315390451844595562142974679, 4.27846002097281827006952561126, 4.42135656551607590398068520042, 4.68399406015828843288797991777, 4.79467338124890664422881234876, 5.04327689127611963480938657351, 5.37604648533010097552585100069, 5.38033739547577080378556808304, 5.91962011969041426407445239974, 5.94018485318890720140628828699, 6.13654817495053243056019956516, 6.15693197229532623232883241238, 6.41679944726593406557584588540, 6.68582092825434791126346096798
Plot not available for L-functions of degree greater than 10.
