| L(s) = 1 | + 2·2-s − 4-s − 5·5-s − 2·8-s − 10·10-s − 2·11-s − 3·13-s + 3·16-s − 12·17-s + 3·19-s + 5·20-s − 4·22-s + 17·25-s − 6·26-s + 29-s − 6·31-s − 2·32-s − 24·34-s + 3·37-s + 6·38-s + 10·40-s − 22·41-s + 3·43-s + 2·44-s + 18·47-s + 34·50-s + 3·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.23·5-s − 0.707·8-s − 3.16·10-s − 0.603·11-s − 0.832·13-s + 3/4·16-s − 2.91·17-s + 0.688·19-s + 1.11·20-s − 0.852·22-s + 17/5·25-s − 1.17·26-s + 0.185·29-s − 1.07·31-s − 0.353·32-s − 4.11·34-s + 0.493·37-s + 0.973·38-s + 1.58·40-s − 3.43·41-s + 0.457·43-s + 0.301·44-s + 2.62·47-s + 4.80·50-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4289214765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4289214765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - T + p T^{2} - 3 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + p T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 62 p T^{7} + 9 p^{2} T^{8} + 7 p^{3} T^{9} + 8 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 416 p T^{7} + 48 p^{2} T^{8} + 34 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 + 12 T + 54 T^{2} + 210 T^{3} + 1350 T^{4} + 5898 T^{5} + 19735 T^{6} + 5898 p T^{7} + 1350 p^{2} T^{8} + 210 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 3 T - 42 T^{2} + 61 T^{3} + 69 p T^{4} - 726 T^{5} - 27501 T^{6} - 726 p T^{7} + 69 p^{3} T^{8} + 61 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 324 p T^{7} + 468 p^{2} T^{8} + 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 3 T + 69 T^{2} + 213 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 + 18 T + 90 T^{2} + 378 T^{3} + 7848 T^{4} + 52668 T^{5} + 160459 T^{6} + 52668 p T^{7} + 7848 p^{2} T^{8} + 378 p^{3} T^{9} + 90 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 9 T + 171 T^{2} - 999 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 6 T + 162 T^{2} + 665 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 3 T - 42 T^{2} - 1209 T^{3} - 3165 T^{4} + 28380 T^{5} + 1003961 T^{6} + 28380 p T^{7} - 3165 p^{2} T^{8} - 1209 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 2 T - 112 T^{2} - 1238 T^{3} + 1662 T^{4} + 59806 T^{5} + 720895 T^{6} + 59806 p T^{7} + 1662 p^{2} T^{8} - 1238 p^{3} T^{9} - 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.87446421233761338316934978290, −4.79026599162461062708780858101, −4.76241308872283839482829837903, −4.54026875418306292203550520744, −4.46659912749071033888976749397, −4.43416078737319559841065809327, −4.14900183809038580480229645265, −3.97201955753257460805180790472, −3.84576798211450343629516951056, −3.76677788351848322213452785857, −3.60117112652933636439567058269, −3.25652750934308741937052952932, −3.12197714873859518006820076239, −2.99636386068624116479470122842, −2.97905561823466975471435164007, −2.70234917462913649689361468449, −2.12268439234079227724664535969, −2.11872202792728077156804613900, −2.04672840930376178749082678989, −1.95408864980247837229920486026, −1.40815270687679638559048349687, −1.06325820134013949483491311932, −0.865938362690973310185825822820, −0.33310268182186156997166355103, −0.16271053730768304552844842518,
0.16271053730768304552844842518, 0.33310268182186156997166355103, 0.865938362690973310185825822820, 1.06325820134013949483491311932, 1.40815270687679638559048349687, 1.95408864980247837229920486026, 2.04672840930376178749082678989, 2.11872202792728077156804613900, 2.12268439234079227724664535969, 2.70234917462913649689361468449, 2.97905561823466975471435164007, 2.99636386068624116479470122842, 3.12197714873859518006820076239, 3.25652750934308741937052952932, 3.60117112652933636439567058269, 3.76677788351848322213452785857, 3.84576798211450343629516951056, 3.97201955753257460805180790472, 4.14900183809038580480229645265, 4.43416078737319559841065809327, 4.46659912749071033888976749397, 4.54026875418306292203550520744, 4.76241308872283839482829837903, 4.79026599162461062708780858101, 4.87446421233761338316934978290
Plot not available for L-functions of degree greater than 10.
