| L(s) = 1 | − 2·2-s + 3·4-s + 3·7-s − 4·8-s − 8·11-s − 6·14-s + 8·16-s + 4·17-s + 7·19-s + 16·22-s + 9·23-s − 4·25-s + 9·28-s − 7·29-s − 14·31-s − 11·32-s − 8·34-s − 14·38-s + 2·41-s + 19·43-s − 24·44-s − 18·46-s + 34·47-s + 3·49-s + 8·50-s − 26·53-s − 12·56-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.13·7-s − 1.41·8-s − 2.41·11-s − 1.60·14-s + 2·16-s + 0.970·17-s + 1.60·19-s + 3.41·22-s + 1.87·23-s − 4/5·25-s + 1.70·28-s − 1.29·29-s − 2.51·31-s − 1.94·32-s − 1.37·34-s − 2.27·38-s + 0.312·41-s + 2.89·43-s − 3.61·44-s − 2.65·46-s + 4.95·47-s + 3/7·49-s + 1.13·50-s − 3.57·53-s − 1.60·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2504213849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2504213849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T + T^{2} )^{3} \) |
| 13 | \( 1 + 5 p T^{3} + p^{3} T^{6} \) |
| good | 2 | \( 1 + p T + T^{2} - 3 T^{4} - 7 T^{5} - 9 T^{6} - 7 p T^{7} - 3 p^{2} T^{8} + p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 5 | \( ( 1 + 2 T^{2} - 13 T^{3} + 2 p T^{4} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 8 T + 14 T^{2} + 38 T^{3} + 502 T^{4} + 1218 T^{5} + 179 T^{6} + 1218 p T^{7} + 502 p^{2} T^{8} + 38 p^{3} T^{9} + 14 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 4 T - 36 T^{2} + 62 T^{3} + 1280 T^{4} - 1104 T^{5} - 22541 T^{6} - 1104 p T^{7} + 1280 p^{2} T^{8} + 62 p^{3} T^{9} - 36 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 7 T - 7 T^{2} + 46 T^{3} + 539 T^{4} + 105 T^{5} - 15594 T^{6} + 105 p T^{7} + 539 p^{2} T^{8} + 46 p^{3} T^{9} - 7 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 9 T - 2 T^{2} + 83 T^{3} + 1245 T^{4} - 4844 T^{5} - 4737 T^{6} - 4844 p T^{7} + 1245 p^{2} T^{8} + 83 p^{3} T^{9} - 2 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 7 T - 24 T^{2} - 311 T^{3} + 335 T^{4} + 6252 T^{5} + 21949 T^{6} + 6252 p T^{7} + 335 p^{2} T^{8} - 311 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 7 T + 53 T^{2} + 153 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 98 T^{2} - 26 T^{3} + 5978 T^{4} + 1274 T^{5} - 253873 T^{6} + 1274 p T^{7} + 5978 p^{2} T^{8} - 26 p^{3} T^{9} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 2 T - 51 T^{2} - 182 T^{3} + 842 T^{4} + 7638 T^{5} + 8389 T^{6} + 7638 p T^{7} + 842 p^{2} T^{8} - 182 p^{3} T^{9} - 51 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 19 T + 116 T^{2} - 929 T^{3} + 14093 T^{4} - 91236 T^{5} + 381219 T^{6} - 91236 p T^{7} + 14093 p^{2} T^{8} - 929 p^{3} T^{9} + 116 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 17 T + 155 T^{2} - 1051 T^{3} + 155 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 13 T + 198 T^{2} + 1365 T^{3} + 198 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 3 T - 158 T^{2} - 157 T^{3} + 17049 T^{4} + 6268 T^{5} - 1157781 T^{6} + 6268 p T^{7} + 17049 p^{2} T^{8} - 157 p^{3} T^{9} - 158 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 13 T - 53 T^{2} - 260 T^{3} + 15293 T^{4} + 45487 T^{5} - 706882 T^{6} + 45487 p T^{7} + 15293 p^{2} T^{8} - 260 p^{3} T^{9} - 53 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 5 T - 102 T^{2} + 85 T^{3} + 6245 T^{4} - 27720 T^{5} - 515717 T^{6} - 27720 p T^{7} + 6245 p^{2} T^{8} + 85 p^{3} T^{9} - 102 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 8 T - 36 T^{2} + 442 T^{3} - 2068 T^{4} + 11172 T^{5} + 68839 T^{6} + 11172 p T^{7} - 2068 p^{2} T^{8} + 442 p^{3} T^{9} - 36 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 2 T + 138 T^{2} + 5 p T^{3} + 138 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + T - 53 T^{2} - 179 T^{3} - 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 2 T + 64 T^{2} + 561 T^{3} + 64 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 19 T + 95 T^{2} + 328 T^{3} + 2185 T^{4} - 120555 T^{5} - 2163058 T^{6} - 120555 p T^{7} + 2185 p^{2} T^{8} + 328 p^{3} T^{9} + 95 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 27 T + 208 T^{2} + 2393 T^{3} + 63305 T^{4} + 566122 T^{5} + 2706977 T^{6} + 566122 p T^{7} + 63305 p^{2} T^{8} + 2393 p^{3} T^{9} + 208 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.53462259184677514671361324627, −5.49507480188178970236311098571, −5.19462372065307857422528263225, −5.15914489142836206311254408116, −4.92311083969833359556021391684, −4.64863124179717388786171034863, −4.37711022540227215104594372200, −4.21333809033189821207273493965, −3.94446422800072620836887025855, −3.92334905264303878528208156846, −3.89108994644129158735089338244, −3.43584852617104891801103962094, −3.16034963315546984836188995231, −2.85667376116803352916420187894, −2.85661752914604382586139877206, −2.77877408945983036602197488176, −2.72683131341486359002059884524, −2.22631395358086750511450063698, −2.09646019832130152317869623911, −1.66829480928252681761851912434, −1.53023243789576804309151795350, −1.36865416409567737178647772933, −1.00130748461816562675267790374, −0.800529732327080538998949357430, −0.11570016734611527429738049951,
0.11570016734611527429738049951, 0.800529732327080538998949357430, 1.00130748461816562675267790374, 1.36865416409567737178647772933, 1.53023243789576804309151795350, 1.66829480928252681761851912434, 2.09646019832130152317869623911, 2.22631395358086750511450063698, 2.72683131341486359002059884524, 2.77877408945983036602197488176, 2.85661752914604382586139877206, 2.85667376116803352916420187894, 3.16034963315546984836188995231, 3.43584852617104891801103962094, 3.89108994644129158735089338244, 3.92334905264303878528208156846, 3.94446422800072620836887025855, 4.21333809033189821207273493965, 4.37711022540227215104594372200, 4.64863124179717388786171034863, 4.92311083969833359556021391684, 5.15914489142836206311254408116, 5.19462372065307857422528263225, 5.49507480188178970236311098571, 5.53462259184677514671361324627
Plot not available for L-functions of degree greater than 10.
