| L(s) = 1 | − 3·4-s + 4·9-s − 12·11-s + 6·16-s + 4·19-s − 8·25-s + 4·29-s − 24·31-s − 12·36-s − 4·41-s + 36·44-s + 8·49-s + 4·59-s + 20·61-s − 10·64-s + 16·71-s − 12·76-s − 56·79-s − 4·89-s − 48·99-s + 24·100-s − 28·101-s + 48·109-s − 12·116-s + 18·121-s + 72·124-s − 4·125-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 4/3·9-s − 3.61·11-s + 3/2·16-s + 0.917·19-s − 8/5·25-s + 0.742·29-s − 4.31·31-s − 2·36-s − 0.624·41-s + 5.42·44-s + 8/7·49-s + 0.520·59-s + 2.56·61-s − 5/4·64-s + 1.89·71-s − 1.37·76-s − 6.30·79-s − 0.423·89-s − 4.82·99-s + 12/5·100-s − 2.78·101-s + 4.59·109-s − 1.11·116-s + 1.63·121-s + 6.46·124-s − 0.357·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1131712103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1131712103\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 8 T^{2} + 4 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 8 T^{2} + 88 T^{4} - 734 T^{6} + 88 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 17 | \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 13 T^{2} - 116 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 70 T^{2} + 2127 T^{4} - 47860 T^{6} + 2127 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 12 T + 113 T^{2} + 664 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 160 T^{2} + 11392 T^{4} - 506230 T^{6} + 11392 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 84 T^{2} + 5136 T^{4} - 202462 T^{6} + 5136 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 216 T^{2} + 21176 T^{4} - 1243838 T^{6} + 21176 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 250 T^{2} + 28167 T^{4} - 1878700 T^{6} + 28167 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 133 T^{2} - 276 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 298 T^{2} + 41783 T^{4} - 3518604 T^{6} + 41783 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 178 T^{2} - 936 T^{3} + 178 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 + 28 T + 453 T^{2} + 4744 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 322 T^{2} + 51623 T^{4} - 5250876 T^{6} + 51623 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 223 T^{2} + 396 T^{3} + 223 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 10 T^{2} + 4687 T^{4} - 11980 T^{6} + 4687 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.84089813890087779243335022612, −4.80143269876903844644337905766, −4.77802347641348181832854879769, −4.34182164446629490977940824207, −4.23211410913785260343198781683, −4.00375827052342296479312051294, −3.87781601760948774511697441767, −3.87326916541307742378792464484, −3.82427051029263443529688038497, −3.69953384113011806359785009108, −3.14602098762121065812877992956, −3.04549089364469482585501851863, −3.02795771818972747266364968732, −2.78070148341416103471960297050, −2.73834397609305040470025598667, −2.26563460952745448349214025586, −2.19665843628802814314671928669, −2.11377932089941415207015971715, −1.81027178097772104989131952507, −1.53881020815766279284031490392, −1.33129671267399843581428842648, −1.19682268467266008988831935260, −0.70934924769663776858458269594, −0.34914101344907292780115050843, −0.087069049613404305233188994865,
0.087069049613404305233188994865, 0.34914101344907292780115050843, 0.70934924769663776858458269594, 1.19682268467266008988831935260, 1.33129671267399843581428842648, 1.53881020815766279284031490392, 1.81027178097772104989131952507, 2.11377932089941415207015971715, 2.19665843628802814314671928669, 2.26563460952745448349214025586, 2.73834397609305040470025598667, 2.78070148341416103471960297050, 3.02795771818972747266364968732, 3.04549089364469482585501851863, 3.14602098762121065812877992956, 3.69953384113011806359785009108, 3.82427051029263443529688038497, 3.87326916541307742378792464484, 3.87781601760948774511697441767, 4.00375827052342296479312051294, 4.23211410913785260343198781683, 4.34182164446629490977940824207, 4.77802347641348181832854879769, 4.80143269876903844644337905766, 4.84089813890087779243335022612
Plot not available for L-functions of degree greater than 10.
