L(s) = 1 | − 2-s − 3-s + 6-s + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 6-s + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2463203317\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2463203317\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 3 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 167 | \( ( 1 - T )^{10} \) |
good | 5 | \( ( 1 + T^{2} )^{10} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) |
| 13 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 17 | \( ( 1 + T^{2} )^{10} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 23 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 37 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 41 | \( ( 1 + T^{2} )^{10} \) |
| 43 | \( ( 1 + T^{2} )^{10} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) |
| 53 | \( ( 1 + T^{2} )^{10} \) |
| 59 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) |
| 67 | \( ( 1 + T^{2} )^{10} \) |
| 71 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 73 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 79 | \( ( 1 + T^{2} )^{10} \) |
| 83 | \( ( 1 - T )^{10}( 1 + T )^{10} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.43062018570299521503068427378, −3.37295572007839185449154365206, −3.33141579554695685553270990557, −3.19567922589099793609408758770, −3.14691847581597375063419529901, −3.03006468940317916439329839449, −2.91223632622808443979817457861, −2.79893431239952689024205794906, −2.42271863402911549332038774359, −2.40030282703516699244742858961, −2.39591821652881302389623929964, −2.25496321622871098715489189879, −2.15659483998477071812008552309, −2.07722669255879354796332111870, −1.77534703422381633789423465072, −1.76741432591720677145242629113, −1.74375930733563195692064760898, −1.67234156056442930718178238898, −1.58219327958329133255244849674, −1.47650705043448272081151416953, −1.19323653369796004732161807247, −0.821356546807012871468183666687, −0.71142446747013975391390430563, −0.48935273346476753254294368942, −0.44030265948757455079775548392,
0.44030265948757455079775548392, 0.48935273346476753254294368942, 0.71142446747013975391390430563, 0.821356546807012871468183666687, 1.19323653369796004732161807247, 1.47650705043448272081151416953, 1.58219327958329133255244849674, 1.67234156056442930718178238898, 1.74375930733563195692064760898, 1.76741432591720677145242629113, 1.77534703422381633789423465072, 2.07722669255879354796332111870, 2.15659483998477071812008552309, 2.25496321622871098715489189879, 2.39591821652881302389623929964, 2.40030282703516699244742858961, 2.42271863402911549332038774359, 2.79893431239952689024205794906, 2.91223632622808443979817457861, 3.03006468940317916439329839449, 3.14691847581597375063419529901, 3.19567922589099793609408758770, 3.33141579554695685553270990557, 3.37295572007839185449154365206, 3.43062018570299521503068427378
Plot not available for L-functions of degree greater than 10.