L(s) = 1 | + 4-s − 2·5-s + 2·9-s − 2·20-s + 25-s − 2·29-s + 2·36-s − 4·45-s − 5·49-s + 81-s + 100-s − 14·101-s + 4·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
L(s) = 1 | + 4-s − 2·5-s + 2·9-s − 2·20-s + 25-s − 2·29-s + 2·36-s − 4·45-s − 5·49-s + 81-s + 100-s − 14·101-s + 4·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1901660827\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1901660827\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
good | 3 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 17 | \( ( 1 + T^{2} )^{12} \) |
| 19 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 23 | \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \) |
| 31 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 37 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 59 | \( ( 1 - T )^{12}( 1 + T )^{12} \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.87338466665928088348578129009, −3.76204784907551265284541038154, −3.59158873308908596442473987257, −3.43768553696380618248968706969, −3.37875345614187953209404857333, −3.35298813573089313724991472813, −3.21768456176988048915957176451, −3.20876723768813770870134505877, −2.98054003384411407087009633286, −2.97941860969455483416574601799, −2.78687499029036030319888381172, −2.62945750008691529249640735255, −2.60935201895516516126512201721, −2.45145430766544914384929883387, −2.28873060754101138061335095651, −2.09374550862002418250966705015, −2.06680096340152300433337871802, −1.80439613628457985677589988269, −1.69893447480238512568315709935, −1.68327825381533313832490027152, −1.61810811584824145909742764500, −1.34909750926378692741777822648, −1.31532271764520104370343220137, −1.10609730403724994252104050984, −0.68914589525004690176147819985,
0.68914589525004690176147819985, 1.10609730403724994252104050984, 1.31532271764520104370343220137, 1.34909750926378692741777822648, 1.61810811584824145909742764500, 1.68327825381533313832490027152, 1.69893447480238512568315709935, 1.80439613628457985677589988269, 2.06680096340152300433337871802, 2.09374550862002418250966705015, 2.28873060754101138061335095651, 2.45145430766544914384929883387, 2.60935201895516516126512201721, 2.62945750008691529249640735255, 2.78687499029036030319888381172, 2.97941860969455483416574601799, 2.98054003384411407087009633286, 3.20876723768813770870134505877, 3.21768456176988048915957176451, 3.35298813573089313724991472813, 3.37875345614187953209404857333, 3.43768553696380618248968706969, 3.59158873308908596442473987257, 3.76204784907551265284541038154, 3.87338466665928088348578129009
Plot not available for L-functions of degree greater than 10.