L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·9-s − 2·12-s + 4·18-s − 2·19-s + 2·25-s − 2·27-s + 2·36-s − 4·38-s + 2·41-s + 2·43-s + 4·50-s − 4·54-s + 4·57-s − 2·59-s − 4·75-s − 2·76-s + 81-s + 4·82-s + 2·83-s + 4·86-s + 2·89-s − 4·97-s + 2·100-s + ⋯ |
L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·9-s − 2·12-s + 4·18-s − 2·19-s + 2·25-s − 2·27-s + 2·36-s − 4·38-s + 2·41-s + 2·43-s + 4·50-s − 4·54-s + 4·57-s − 2·59-s − 4·75-s − 2·76-s + 81-s + 4·82-s + 2·83-s + 4·86-s + 2·89-s − 4·97-s + 2·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1595745096\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1595745096\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) |
good | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 5 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 7 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 11 | \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 13 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 17 | \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 19 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 23 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 29 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 31 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 37 | \( ( 1 + T^{4} )^{10} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 59 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 61 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.67780527048255904571806746803, −2.64240277343721040094180467435, −2.47807395129496890565234716984, −2.46265067207093483053294352210, −2.44044882206428685175121067365, −2.27520708070380661986929873867, −2.26117913916977559249081262239, −2.20654960844176425300557546976, −2.19741629301470957274176104910, −2.19534363424892827770481020801, −2.19075390156466861190449203799, −2.04731181142751525673010614761, −1.87780954807291510019875714084, −1.61745954927173754872982462118, −1.56677321841894246317455430573, −1.53958419053261519134445723691, −1.45395325214318122294701810424, −1.39761636828549573603735685266, −1.28365033677039488142098374428, −1.28094185708622593531553461456, −1.15124353710273176668984306261, −1.06365130455925400985305775541, −1.02009823127570854373240134823, −0.802986117586877474710365817099, −0.60152963256656413118709655715,
0.60152963256656413118709655715, 0.802986117586877474710365817099, 1.02009823127570854373240134823, 1.06365130455925400985305775541, 1.15124353710273176668984306261, 1.28094185708622593531553461456, 1.28365033677039488142098374428, 1.39761636828549573603735685266, 1.45395325214318122294701810424, 1.53958419053261519134445723691, 1.56677321841894246317455430573, 1.61745954927173754872982462118, 1.87780954807291510019875714084, 2.04731181142751525673010614761, 2.19075390156466861190449203799, 2.19534363424892827770481020801, 2.19741629301470957274176104910, 2.20654960844176425300557546976, 2.26117913916977559249081262239, 2.27520708070380661986929873867, 2.44044882206428685175121067365, 2.46265067207093483053294352210, 2.47807395129496890565234716984, 2.64240277343721040094180467435, 2.67780527048255904571806746803
Plot not available for L-functions of degree greater than 10.