L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 2·8-s − 2·9-s − 11-s + 2·14-s + 16-s − 4·18-s − 2·22-s − 23-s − 2·25-s + 3·28-s − 29-s − 6·36-s − 37-s − 9·43-s − 3·44-s − 2·46-s + 49-s − 4·50-s + 2·53-s + 2·56-s − 2·58-s − 2·63-s + 67-s − 71-s + ⋯ |
L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 2·8-s − 2·9-s − 11-s + 2·14-s + 16-s − 4·18-s − 2·22-s − 23-s − 2·25-s + 3·28-s − 29-s − 6·36-s − 37-s − 9·43-s − 3·44-s − 2·46-s + 49-s − 4·50-s + 2·53-s + 2·56-s − 2·58-s − 2·63-s + 67-s − 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09602292915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09602292915\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
| 67 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
good | 2 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 11 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 37 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 43 | \( ( 1 + T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 97 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
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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
−2.94889826530905127526647405999, −2.94771658431199418949910222521, −2.85021183903386048119490560132, −2.83162428020798949859095837238, −2.72540799459410739377025855493, −2.58061037285104260090156658034, −2.52074339906064103294967114253, −2.39695158864687070922694626565, −2.38347050101406084652669767212, −2.29744078856881701565896449287, −2.24635860648660622821238047035, −2.19395926388987145533005302223, −2.03358300396673131607422109053, −2.00959806934931622663597479177, −1.93060646115033413893794571215, −1.86294339093555698853766874905, −1.74405819130659089119328917482, −1.71372483739206753914299019160, −1.62743312651104966769738233618, −1.46085677850117055015982965937, −1.35332906324987409453610031097, −1.25280790371751161020699070125, −1.23183718474848882676955753624, −1.16302497302257436246811061647, −0.818933133218357113705426334412,
0.818933133218357113705426334412, 1.16302497302257436246811061647, 1.23183718474848882676955753624, 1.25280790371751161020699070125, 1.35332906324987409453610031097, 1.46085677850117055015982965937, 1.62743312651104966769738233618, 1.71372483739206753914299019160, 1.74405819130659089119328917482, 1.86294339093555698853766874905, 1.93060646115033413893794571215, 2.00959806934931622663597479177, 2.03358300396673131607422109053, 2.19395926388987145533005302223, 2.24635860648660622821238047035, 2.29744078856881701565896449287, 2.38347050101406084652669767212, 2.39695158864687070922694626565, 2.52074339906064103294967114253, 2.58061037285104260090156658034, 2.72540799459410739377025855493, 2.83162428020798949859095837238, 2.85021183903386048119490560132, 2.94771658431199418949910222521, 2.94889826530905127526647405999
Plot not available for L-functions of degree greater than 10.