| L(s) = 1 | + 3-s + 3·5-s + 4·7-s + 9-s + 10·11-s + 3·13-s + 3·15-s + 3·17-s + 4·21-s + 14·23-s + 3·25-s − 4·27-s − 6·29-s + 22·31-s + 10·33-s + 12·35-s + 4·37-s + 3·39-s − 6·41-s + 5·43-s + 3·45-s + 6·47-s − 16·49-s + 3·51-s + 13·53-s + 30·55-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1.51·7-s + 1/3·9-s + 3.01·11-s + 0.832·13-s + 0.774·15-s + 0.727·17-s + 0.872·21-s + 2.91·23-s + 3/5·25-s − 0.769·27-s − 1.11·29-s + 3.95·31-s + 1.74·33-s + 2.02·35-s + 0.657·37-s + 0.480·39-s − 0.937·41-s + 0.762·43-s + 0.447·45-s + 0.875·47-s − 2.28·49-s + 0.420·51-s + 1.78·53-s + 4.04·55-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.46128049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.46128049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{3} \) |
| 19 | \( 1 - 18 T^{2} + 7 T^{3} - 18 p T^{4} + p^{3} T^{6} \) |
| good | 3 | \( 1 - T + 5 T^{3} - 5 T^{4} - 2 p T^{5} + 17 p T^{6} - 2 p^{2} T^{7} - 5 p^{2} T^{8} + 5 p^{3} T^{9} - p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 2 T + 2 p T^{2} - 23 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 5 T + 3 p T^{2} - 101 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 - 3 T - 6 T^{2} - 3 T^{3} - 93 T^{4} + 348 T^{5} + 3841 T^{6} + 348 p T^{7} - 93 p^{2} T^{8} - 3 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 14 T + 88 T^{2} - 314 T^{3} + 794 T^{4} - 2546 T^{5} + 11179 T^{6} - 2546 p T^{7} + 794 p^{2} T^{8} - 314 p^{3} T^{9} + 88 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 6 T + 30 T^{2} + 366 T^{3} + 852 T^{4} + 1968 T^{5} + 44683 T^{6} + 1968 p T^{7} + 852 p^{2} T^{8} + 366 p^{3} T^{9} + 30 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 11 T + 107 T^{2} - 611 T^{3} + 107 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 2 T + 104 T^{2} - 143 T^{3} + 104 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 6 T - 24 T^{2} - 354 T^{3} - 1002 T^{4} + 1986 T^{5} + 57175 T^{6} + 1986 p T^{7} - 1002 p^{2} T^{8} - 354 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 5 T - 42 T^{2} + 735 T^{3} - 1067 T^{4} - 16280 T^{5} + 205987 T^{6} - 16280 p T^{7} - 1067 p^{2} T^{8} + 735 p^{3} T^{9} - 42 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 6 T - 24 T^{2} - 258 T^{3} + 474 T^{4} + 17526 T^{5} - 38477 T^{6} + 17526 p T^{7} + 474 p^{2} T^{8} - 258 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 13 T - 20 T^{2} + 281 T^{3} + 7907 T^{4} - 34816 T^{5} - 196019 T^{6} - 34816 p T^{7} + 7907 p^{2} T^{8} + 281 p^{3} T^{9} - 20 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 6 T - 60 T^{2} - 186 T^{3} + 2382 T^{4} + 28362 T^{5} - 220277 T^{6} + 28362 p T^{7} + 2382 p^{2} T^{8} - 186 p^{3} T^{9} - 60 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 7 T - 117 T^{2} - 420 T^{3} + 11893 T^{4} + 15589 T^{5} - 783602 T^{6} + 15589 p T^{7} + 11893 p^{2} T^{8} - 420 p^{3} T^{9} - 117 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 4 T - 93 T^{2} - 972 T^{3} + 1762 T^{4} + 36268 T^{5} + 229291 T^{6} + 36268 p T^{7} + 1762 p^{2} T^{8} - 972 p^{3} T^{9} - 93 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 9 T - 114 T^{2} + 747 T^{3} + 12921 T^{4} - 40104 T^{5} - 827273 T^{6} - 40104 p T^{7} + 12921 p^{2} T^{8} + 747 p^{3} T^{9} - 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 18 T + 36 T^{2} + 122 T^{3} + 14166 T^{4} - 103122 T^{5} + 34935 T^{6} - 103122 p T^{7} + 14166 p^{2} T^{8} + 122 p^{3} T^{9} + 36 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 32 T + 479 T^{2} + 5616 T^{3} + 64786 T^{4} + 666232 T^{5} + 6082063 T^{6} + 666232 p T^{7} + 64786 p^{2} T^{8} + 5616 p^{3} T^{9} + 479 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 31 T + 543 T^{2} + 6001 T^{3} + 543 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 2 T - 131 T^{2} - 298 T^{3} + 5642 T^{4} + 6122 T^{5} - 163115 T^{6} + 6122 p T^{7} + 5642 p^{2} T^{8} - 298 p^{3} T^{9} - 131 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 11 T - 76 T^{2} - 1491 T^{3} - 419 T^{4} + 43414 T^{5} + 267625 T^{6} + 43414 p T^{7} - 419 p^{2} T^{8} - 1491 p^{3} T^{9} - 76 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.48193087252934217349618102331, −5.79809858957910554948115090876, −5.62524471257861957181020672391, −5.58496707864959900837781887009, −5.49946840747898845250867452852, −5.49595126821813442174649274860, −5.27365491749742548685100334404, −4.59576026771695193083575584189, −4.55055626786396321198400741879, −4.45555704964659575626871601899, −4.35574261326355407857798510147, −4.23063384828580331039131239417, −4.04839693833413444013890709896, −3.52149807909189251944083167807, −3.37370794355647503867387934307, −3.34645317927965340497117211025, −2.90526277020918222790906670142, −2.74807458704823105042353756459, −2.51902408342793371346145184603, −2.32405093881321929532957298535, −1.66428680258569218101136840898, −1.56030245347418854257566038805, −1.39284869385322592627459890052, −1.12327458086508999284962222690, −1.07924531106351121287780469930,
1.07924531106351121287780469930, 1.12327458086508999284962222690, 1.39284869385322592627459890052, 1.56030245347418854257566038805, 1.66428680258569218101136840898, 2.32405093881321929532957298535, 2.51902408342793371346145184603, 2.74807458704823105042353756459, 2.90526277020918222790906670142, 3.34645317927965340497117211025, 3.37370794355647503867387934307, 3.52149807909189251944083167807, 4.04839693833413444013890709896, 4.23063384828580331039131239417, 4.35574261326355407857798510147, 4.45555704964659575626871601899, 4.55055626786396321198400741879, 4.59576026771695193083575584189, 5.27365491749742548685100334404, 5.49595126821813442174649274860, 5.49946840747898845250867452852, 5.58496707864959900837781887009, 5.62524471257861957181020672391, 5.79809858957910554948115090876, 6.48193087252934217349618102331
Plot not available for L-functions of degree greater than 10.
