| L(s) = 1 | − 24·19-s − 3·25-s + 4·31-s − 4·49-s − 28·61-s + 48·79-s − 112·109-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.50·19-s − 3/5·25-s + 0.718·31-s − 4/7·49-s − 3.58·61-s + 5.40·79-s − 10.7·109-s + 3.81·121-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 − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02905738905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02905738905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 3 T^{2} - 11 T^{4} - 18 p T^{6} - 18 p^{2} T^{8} - 18 p^{3} T^{10} - 11 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \) |
| good | 7 | \( ( 1 + 2 T^{2} + 34 T^{4} - 256 T^{6} - 1697 T^{8} - 256 p^{2} T^{10} + 34 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{4}( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} - 47 T^{4} - 290 T^{6} - 26426 T^{8} - 290 p^{2} T^{10} - 47 p^{4} T^{12} + p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 15 T^{2} + 602 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{8} \) |
| 23 | \( ( 1 + 24 T^{2} - 110 T^{4} - 8928 T^{6} - 113949 T^{8} - 8928 p^{2} T^{10} - 110 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 31 T^{2} + 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - T - 29 T^{2} + 32 T^{3} - 92 T^{4} + 32 p T^{5} - 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 59 T^{2} + 3318 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 129 T^{2} + 9409 T^{4} - 499230 T^{6} + 21502542 T^{8} - 499230 p^{2} T^{10} + 9409 p^{4} T^{12} - 129 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 + 146 T^{2} + 12418 T^{4} + 759200 T^{6} + 36342319 T^{8} + 759200 p^{2} T^{10} + 12418 p^{4} T^{12} + 146 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 + 18 T^{2} - 950 T^{4} - 56592 T^{6} - 3972321 T^{8} - 56592 p^{2} T^{10} - 950 p^{4} T^{12} + 18 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 134 T^{2} + 8946 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 3 p T^{2} + 17341 T^{4} - 21078 p T^{6} + 74490534 T^{8} - 21078 p^{3} T^{10} + 17341 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 7 T - 53 T^{2} - 140 T^{3} + 3694 T^{4} - 140 p T^{5} - 53 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 8 T^{2} + 3970 T^{4} - 103072 T^{6} - 5181581 T^{8} - 103072 p^{2} T^{10} + 3970 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 53 T^{2} + 3528 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 227 T^{2} + 22734 T^{4} - 227 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 + 210 T^{2} + 20458 T^{4} + 2071440 T^{6} + 207680943 T^{8} + 2071440 p^{2} T^{10} + 20458 p^{4} T^{12} + 210 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 246 T^{2} + 28907 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 182 T^{2} + 16474 T^{4} - 394576 T^{6} - 122970497 T^{8} - 394576 p^{2} T^{10} + 16474 p^{4} T^{12} + 182 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32742816314195375237471773992, −2.24625598883564968524010604550, −2.19362699795057648520404261815, −2.08500923067404500357886931116, −2.04604359561509017836968457008, −2.04386572690435973831271404783, −1.96872348978119078149566300262, −1.95117638423494783475785305635, −1.82315386146693873733960519861, −1.64446217330760688797560269864, −1.64315456442881756756998308671, −1.62787322620764174368517066621, −1.36006580129881154618013618161, −1.30074748956038411834469220629, −1.24636225140191034768854084412, −1.22042631456308810225506503996, −1.07773322914300897378134649054, −0.939709535878905459624224337833, −0.881036648444606074505320985311, −0.72689406173775826664261221204, −0.55323881857158931347194670846, −0.45684484790436249001740010732, −0.23843110766605416678379020671, −0.18676377983837126793547797944, −0.01959595748559724578333426395,
0.01959595748559724578333426395, 0.18676377983837126793547797944, 0.23843110766605416678379020671, 0.45684484790436249001740010732, 0.55323881857158931347194670846, 0.72689406173775826664261221204, 0.881036648444606074505320985311, 0.939709535878905459624224337833, 1.07773322914300897378134649054, 1.22042631456308810225506503996, 1.24636225140191034768854084412, 1.30074748956038411834469220629, 1.36006580129881154618013618161, 1.62787322620764174368517066621, 1.64315456442881756756998308671, 1.64446217330760688797560269864, 1.82315386146693873733960519861, 1.95117638423494783475785305635, 1.96872348978119078149566300262, 2.04386572690435973831271404783, 2.04604359561509017836968457008, 2.08500923067404500357886931116, 2.19362699795057648520404261815, 2.24625598883564968524010604550, 2.32742816314195375237471773992
Plot not available for L-functions of degree greater than 10.
