| L(s) = 1 | − 3·4-s + 2·5-s + 10·9-s − 12·11-s + 6·16-s + 34·19-s − 6·20-s + 25-s + 10·29-s − 18·31-s − 30·36-s − 10·41-s + 36·44-s + 20·45-s + 31·49-s − 24·55-s − 40·59-s − 42·61-s − 10·64-s − 16·71-s − 102·76-s + 18·79-s + 12·80-s + 49·81-s + 28·89-s + 68·95-s − 120·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s + 10/3·9-s − 3.61·11-s + 3/2·16-s + 7.80·19-s − 1.34·20-s + 1/5·25-s + 1.85·29-s − 3.23·31-s − 5·36-s − 1.56·41-s + 5.42·44-s + 2.98·45-s + 31/7·49-s − 3.23·55-s − 5.20·59-s − 5.37·61-s − 5/4·64-s − 1.89·71-s − 11.7·76-s + 2.02·79-s + 1.34·80-s + 49/9·81-s + 2.96·89-s + 6.97·95-s − 12.0·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.025512468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025512468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - 10 T^{2} + 17 p T^{4} - 176 T^{6} + 17 p^{3} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 31 T^{2} + 454 T^{4} - 4003 T^{6} + 454 p^{2} T^{8} - 31 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 6 T + 41 T^{2} + 130 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 43 T^{2} + 878 T^{4} - 12687 T^{6} + 878 p^{2} T^{8} - 43 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 42 T^{2} + 1155 T^{4} - 24816 T^{6} + 1155 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 17 T + 148 T^{2} - 797 T^{3} + 148 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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 | \( ( 1 - 5 T + 82 T^{2} - 291 T^{3} + 82 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 9 T + 84 T^{2} + 423 T^{3} + 84 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 82 T^{2} + 4015 T^{4} - 159016 T^{6} + 4015 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 5 T + 46 T^{2} + 225 T^{3} + 46 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 170 T^{2} + 15279 T^{4} - 888968 T^{6} + 15279 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 138 T^{2} + 14295 T^{4} - 854556 T^{6} + 14295 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 20 T + 249 T^{2} + 2088 T^{3} + 249 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 21 T + 272 T^{2} + 2329 T^{3} + 272 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 87 T^{2} + 978 T^{4} + 259189 T^{6} + 978 p^{2} T^{8} - 87 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 159 T^{2} + 930 T^{3} + 159 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 51 T^{2} + 5858 T^{4} - 385823 T^{6} + 5858 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 9 T + 234 T^{2} - 1361 T^{3} + 234 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 34 T^{2} + 93 p T^{4} + 448204 T^{6} + 93 p^{3} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T - T^{2} + 1286 T^{3} - p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 510 T^{2} + 114171 T^{4} - 14374424 T^{6} + 114171 p^{2} T^{8} - 510 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.86262504894252141220708906027, −5.67635285852787839968425765659, −5.53232288033794923834525413611, −5.47601049507832333045973775667, −5.33331422339612600212546287453, −5.04704483221589622358828710383, −5.04190850683663902689001361114, −4.80612364862519583032037581322, −4.73174050959866337808584633621, −4.54639417093841164204066079301, −4.33903469775137551110217113012, −3.91566442145129655382118194903, −3.72166524912093386471444064103, −3.67828743465090292793877691644, −3.18237198796517343579781034719, −3.11379426852302089219060232602, −2.92068540425699385286658594211, −2.87340324050520458732078610333, −2.74723234378950691606945550297, −1.88561150356206937283539584660, −1.75021217101944983312230311255, −1.60067863146845453800630682691, −1.15821843740373829515087899277, −1.09823879401322664863352925075, −0.49564602043173311008570106138,
0.49564602043173311008570106138, 1.09823879401322664863352925075, 1.15821843740373829515087899277, 1.60067863146845453800630682691, 1.75021217101944983312230311255, 1.88561150356206937283539584660, 2.74723234378950691606945550297, 2.87340324050520458732078610333, 2.92068540425699385286658594211, 3.11379426852302089219060232602, 3.18237198796517343579781034719, 3.67828743465090292793877691644, 3.72166524912093386471444064103, 3.91566442145129655382118194903, 4.33903469775137551110217113012, 4.54639417093841164204066079301, 4.73174050959866337808584633621, 4.80612364862519583032037581322, 5.04190850683663902689001361114, 5.04704483221589622358828710383, 5.33331422339612600212546287453, 5.47601049507832333045973775667, 5.53232288033794923834525413611, 5.67635285852787839968425765659, 5.86262504894252141220708906027
Plot not available for L-functions of degree greater than 10.
