L(s) = 1 | − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s + 5·29-s + 2·34-s − 2·37-s + 5·41-s + 2·45-s − 49-s − 50-s − 2·53-s − 5·58-s + 5·61-s + 4·65-s − 2·73-s + 2·74-s − 5·82-s + 4·85-s − 2·89-s − 2·90-s + 5·97-s + ⋯ |
L(s) = 1 | − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s + 5·29-s + 2·34-s − 2·37-s + 5·41-s + 2·45-s − 49-s − 50-s − 2·53-s − 5·58-s + 5·61-s + 4·65-s − 2·73-s + 2·74-s − 5·82-s + 4·85-s − 2·89-s − 2·90-s + 5·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 113^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 113^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08561175439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08561175439\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 113 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
good | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 17 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 29 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 37 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 41 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 53 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 61 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 89 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 97 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.37738611531245151470225453459, −6.17613757555279566517240216938, −5.91782516544700143766077160240, −5.67624016081202111093347898546, −5.61190696808264609621861803618, −5.37425679665088421437526603324, −5.11634557821802270049300415849, −4.81398978346492242300991355994, −4.65888305824525755354354072343, −4.58716423304943373434452259585, −4.43751797058563457910429471311, −4.40371012869811016999261777931, −4.17140563303022896733019107648, −3.85152727986216642109319650566, −3.61426906456727793850305807768, −3.35430466451546039977032209457, −3.12007175008936305061275410816, −2.91098535368697768194942361365, −2.81812029407126100919288603052, −2.44669328956368324692310820091, −2.23771845633985248717428959640, −2.18370179988195618287774791319, −1.74499022300838458996852691989, −0.871106512214654045672066458374, −0.76014129851655960264478078975,
0.76014129851655960264478078975, 0.871106512214654045672066458374, 1.74499022300838458996852691989, 2.18370179988195618287774791319, 2.23771845633985248717428959640, 2.44669328956368324692310820091, 2.81812029407126100919288603052, 2.91098535368697768194942361365, 3.12007175008936305061275410816, 3.35430466451546039977032209457, 3.61426906456727793850305807768, 3.85152727986216642109319650566, 4.17140563303022896733019107648, 4.40371012869811016999261777931, 4.43751797058563457910429471311, 4.58716423304943373434452259585, 4.65888305824525755354354072343, 4.81398978346492242300991355994, 5.11634557821802270049300415849, 5.37425679665088421437526603324, 5.61190696808264609621861803618, 5.67624016081202111093347898546, 5.91782516544700143766077160240, 6.17613757555279566517240216938, 6.37738611531245151470225453459
Plot not available for L-functions of degree greater than 10.