| L(s) = 1 | − 3·2-s + 3·4-s + 3·5-s + 6·7-s − 9·10-s + 6·11-s + 6·13-s − 18·14-s − 3·16-s + 9·17-s + 12·19-s + 9·20-s − 18·22-s + 12·23-s − 6·25-s − 18·26-s + 18·28-s − 21·29-s + 15·31-s + 6·32-s − 27·34-s + 18·35-s + 3·37-s − 36·38-s + 12·41-s + 6·43-s + 18·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.34·5-s + 2.26·7-s − 2.84·10-s + 1.80·11-s + 1.66·13-s − 4.81·14-s − 3/4·16-s + 2.18·17-s + 2.75·19-s + 2.01·20-s − 3.83·22-s + 2.50·23-s − 6/5·25-s − 3.53·26-s + 3.40·28-s − 3.89·29-s + 2.69·31-s + 1.06·32-s − 4.63·34-s + 3.04·35-s + 0.493·37-s − 5.83·38-s + 1.87·41-s + 0.914·43-s + 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.799493779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.799493779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + 13 T^{6} + 3 p^{3} T^{7} + 3 p^{4} T^{8} + 9 p^{3} T^{9} + 3 p^{5} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 - 3 T + 3 p T^{2} - 18 T^{3} + 84 T^{4} - 84 T^{5} + 499 T^{6} - 84 p T^{7} + 84 p^{2} T^{8} - 18 p^{3} T^{9} + 3 p^{5} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 6 T + 33 T^{2} - 127 T^{3} + 477 T^{4} - 1485 T^{5} + 4266 T^{6} - 1485 p T^{7} + 477 p^{2} T^{8} - 127 p^{3} T^{9} + 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T + 51 T^{2} - 189 T^{3} + 1065 T^{4} - 3255 T^{5} + 14638 T^{6} - 3255 p T^{7} + 1065 p^{2} T^{8} - 189 p^{3} T^{9} + 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 6 T + 60 T^{2} - 316 T^{3} + 1674 T^{4} - 7362 T^{5} + 27549 T^{6} - 7362 p T^{7} + 1674 p^{2} T^{8} - 316 p^{3} T^{9} + 60 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 9 T + 6 p T^{2} - 630 T^{3} + 4227 T^{4} - 19611 T^{5} + 5591 p T^{6} - 19611 p T^{7} + 4227 p^{2} T^{8} - 630 p^{3} T^{9} + 6 p^{5} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 12 T + 129 T^{2} - 985 T^{3} + 6471 T^{4} - 35073 T^{5} + 166182 T^{6} - 35073 p T^{7} + 6471 p^{2} T^{8} - 985 p^{3} T^{9} + 129 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 12 T + 159 T^{2} - 1179 T^{3} + 9111 T^{4} - 49215 T^{5} + 275674 T^{6} - 49215 p T^{7} + 9111 p^{2} T^{8} - 1179 p^{3} T^{9} + 159 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 21 T + 285 T^{2} + 2826 T^{3} + 23214 T^{4} + 158142 T^{5} + 925699 T^{6} + 158142 p T^{7} + 23214 p^{2} T^{8} + 2826 p^{3} T^{9} + 285 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 15 T + 231 T^{2} - 2098 T^{3} + 18657 T^{4} - 120951 T^{5} + 771318 T^{6} - 120951 p T^{7} + 18657 p^{2} T^{8} - 2098 p^{3} T^{9} + 231 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 3 T + 93 T^{2} - 94 T^{3} + 4608 T^{4} + 864 T^{5} + 171555 T^{6} + 864 p T^{7} + 4608 p^{2} T^{8} - 94 p^{3} T^{9} + 93 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 12 T + 186 T^{2} - 1611 T^{3} + 15249 T^{4} - 110622 T^{5} + 786037 T^{6} - 110622 p T^{7} + 15249 p^{2} T^{8} - 1611 p^{3} T^{9} + 186 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 6 T + 150 T^{2} - 658 T^{3} + 10935 T^{4} - 42948 T^{5} + 562068 T^{6} - 42948 p T^{7} + 10935 p^{2} T^{8} - 658 p^{3} T^{9} + 150 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 15 T + 240 T^{2} - 2466 T^{3} + 24663 T^{4} - 198627 T^{5} + 1483384 T^{6} - 198627 p T^{7} + 24663 p^{2} T^{8} - 2466 p^{3} T^{9} + 240 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 9 T + 237 T^{2} - 1656 T^{3} + 26583 T^{4} - 151479 T^{5} + 1786030 T^{6} - 151479 p T^{7} + 26583 p^{2} T^{8} - 1656 p^{3} T^{9} + 237 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 6 T + 249 T^{2} + 1395 T^{3} + 28119 T^{4} + 143889 T^{5} + 1994854 T^{6} + 143889 p T^{7} + 28119 p^{2} T^{8} + 1395 p^{3} T^{9} + 249 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 24 T + 483 T^{2} - 6769 T^{3} + 81099 T^{4} - 795771 T^{5} + 6755130 T^{6} - 795771 p T^{7} + 81099 p^{2} T^{8} - 6769 p^{3} T^{9} + 483 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 15 T + 348 T^{2} - 4006 T^{3} + 54657 T^{4} - 482175 T^{5} + 4783932 T^{6} - 482175 p T^{7} + 54657 p^{2} T^{8} - 4006 p^{3} T^{9} + 348 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 246 T^{2} + 864 T^{3} + 27087 T^{4} + 163296 T^{5} + 2111380 T^{6} + 163296 p T^{7} + 27087 p^{2} T^{8} + 864 p^{3} T^{9} + 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 12 T + 282 T^{2} - 2326 T^{3} + 34542 T^{4} - 226674 T^{5} + 2873211 T^{6} - 226674 p T^{7} + 34542 p^{2} T^{8} - 2326 p^{3} T^{9} + 282 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 24 T + 501 T^{2} - 6805 T^{3} + 78525 T^{4} - 785493 T^{5} + 7046370 T^{6} - 785493 p T^{7} + 78525 p^{2} T^{8} - 6805 p^{3} T^{9} + 501 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 6 T + 411 T^{2} - 1989 T^{3} + 75405 T^{4} - 3549 p T^{5} + 8002390 T^{6} - 3549 p^{2} T^{7} + 75405 p^{2} T^{8} - 1989 p^{3} T^{9} + 411 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 9 T + 309 T^{2} - 3006 T^{3} + 49110 T^{4} - 463734 T^{5} + 5132383 T^{6} - 463734 p T^{7} + 49110 p^{2} T^{8} - 3006 p^{3} T^{9} + 309 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 21 T + 654 T^{2} + 9026 T^{3} + 158589 T^{4} + 1610523 T^{5} + 20266389 T^{6} + 1610523 p T^{7} + 158589 p^{2} T^{8} + 9026 p^{3} T^{9} + 654 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.58476695851257777719103905831, −5.37843378445568793119819717287, −5.19378731119956203054696392547, −5.19334468670348608286477884622, −4.96916465618776219066087924016, −4.77920327286385818733136360621, −4.76568339855758623637334294215, −4.10284203300566520078979411695, −4.00043091038429119732600511879, −3.99673268960831030580931426632, −3.87763835950422610210172208167, −3.72501173676307956100822876708, −3.40388547951839954470447772326, −3.09116245082537454398632213836, −3.01302537208609964447957061125, −2.86788501803561593153001593501, −2.17405774737837463375929275576, −2.08243058422224331483037034773, −2.08079925344298818040771693188, −1.97151419509781647553660511517, −1.20821513091691800461851737607, −1.11744055592954130864401262748, −1.07623755241472714143105800585, −0.988671212626980202574840408207, −0.829257742670493852461042322898,
0.829257742670493852461042322898, 0.988671212626980202574840408207, 1.07623755241472714143105800585, 1.11744055592954130864401262748, 1.20821513091691800461851737607, 1.97151419509781647553660511517, 2.08079925344298818040771693188, 2.08243058422224331483037034773, 2.17405774737837463375929275576, 2.86788501803561593153001593501, 3.01302537208609964447957061125, 3.09116245082537454398632213836, 3.40388547951839954470447772326, 3.72501173676307956100822876708, 3.87763835950422610210172208167, 3.99673268960831030580931426632, 4.00043091038429119732600511879, 4.10284203300566520078979411695, 4.76568339855758623637334294215, 4.77920327286385818733136360621, 4.96916465618776219066087924016, 5.19334468670348608286477884622, 5.19378731119956203054696392547, 5.37843378445568793119819717287, 5.58476695851257777719103905831
Plot not available for L-functions of degree greater than 10.
