| L(s) = 1 | + 10·9-s + 4·19-s + 25-s + 4·29-s + 16·41-s + 26·49-s + 4·59-s − 28·61-s − 20·71-s + 8·79-s + 45·81-s − 20·89-s − 4·101-s − 6·121-s + 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3·169-s + 40·171-s + 173-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 0.917·19-s + 1/5·25-s + 0.742·29-s + 2.49·41-s + 26/7·49-s + 0.520·59-s − 3.58·61-s − 2.37·71-s + 0.900·79-s + 5·81-s − 2.11·89-s − 0.398·101-s − 0.545·121-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.230·169-s + 3.05·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.104416714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.104416714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - 10 T^{2} + 55 T^{4} - 200 T^{6} + 55 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 26 T^{2} + 319 T^{4} - 2588 T^{6} + 319 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 3 T^{2} + 2 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 42 T^{2} + 815 T^{4} - 13196 T^{6} + 815 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 43 T^{2} - 86 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 118 T^{2} + 6135 T^{4} - 181696 T^{6} + 6135 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 79 T^{2} - 120 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 59 T^{2} - 2 p T^{3} + 59 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 46 T^{2} - 713 T^{4} + 87404 T^{6} - 713 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 8 T + 91 T^{2} - 528 T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 6 T^{2} + 4863 T^{4} - 16288 T^{6} + 4863 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 138 T^{2} + 12047 T^{4} - 662780 T^{6} + 12047 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 94 T^{2} + 7959 T^{4} - 477892 T^{6} + 7959 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 103 T^{2} - 162 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 14 T + 99 T^{2} + 416 T^{3} + 99 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 322 T^{2} + 47495 T^{4} - 4069164 T^{6} + 47495 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 10 T + 187 T^{2} + 1086 T^{3} + 187 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 294 T^{2} + 40479 T^{4} - 3542852 T^{6} + 40479 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + 189 T^{2} - 712 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 106 T^{2} + 4535 T^{4} - 185916 T^{6} + 4535 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 10 T + 55 T^{2} - 276 T^{3} + 55 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 154 T^{2} + 30607 T^{4} - 2814508 T^{6} + 30607 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.18148057631492992678914264010, −5.18042542216919522769258784229, −4.88556011636824526005385449425, −4.53872572181501574439544445815, −4.48026616006020370918930743845, −4.47948906273597738509942890953, −4.16042645830490356780171002718, −4.14246767745518142745702595243, −4.11990660569068740396549424475, −3.94182697644705869292027954750, −3.75038809738704148275207803532, −3.24018784031172692090267654277, −3.04094868655912842928231370325, −2.99420071560379346204693097635, −2.94240334245457577645363798026, −2.87456771409391452315384943953, −2.19121751962343367188112277342, −2.15769294878845106275580060653, −2.02009441178149428630419211969, −1.64557757924857086803746772231, −1.62364621725572143097095855714, −1.13762931404902995065657504552, −0.954498818028575248141131952114, −0.944244681753243944596011630346, −0.43533267742482873847133862995,
0.43533267742482873847133862995, 0.944244681753243944596011630346, 0.954498818028575248141131952114, 1.13762931404902995065657504552, 1.62364621725572143097095855714, 1.64557757924857086803746772231, 2.02009441178149428630419211969, 2.15769294878845106275580060653, 2.19121751962343367188112277342, 2.87456771409391452315384943953, 2.94240334245457577645363798026, 2.99420071560379346204693097635, 3.04094868655912842928231370325, 3.24018784031172692090267654277, 3.75038809738704148275207803532, 3.94182697644705869292027954750, 4.11990660569068740396549424475, 4.14246767745518142745702595243, 4.16042645830490356780171002718, 4.47948906273597738509942890953, 4.48026616006020370918930743845, 4.53872572181501574439544445815, 4.88556011636824526005385449425, 5.18042542216919522769258784229, 5.18148057631492992678914264010
Plot not available for L-functions of degree greater than 10.
