| L(s) = 1 | − 4-s + 2·7-s + 16-s + 25-s − 2·28-s + 3·31-s − 13·37-s − 2·43-s + 49-s + 4·61-s − 67-s − 79-s − 100-s + 3·103-s + 109-s + 2·112-s − 121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s + 13·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s + 16-s + 25-s − 2·28-s + 3·31-s − 13·37-s − 2·43-s + 49-s + 4·61-s − 67-s − 79-s − 100-s + 3·103-s + 109-s + 2·112-s − 121-s − 3·124-s + 127-s + 131-s + 137-s + 139-s + 13·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.004658855201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004658855201\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| good | 2 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 11 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 29 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 37 | \( ( 1 + T )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 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 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 53 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 61 | \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 79 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
| 97 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \) |
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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
−3.31245763798243748049238061128, −3.26294381228667612873956131774, −3.16415375258652697324958799728, −3.01902831265171959151218343778, −2.98790083069800652164428781202, −2.88361662340205635903268130157, −2.72279859696078191042970372981, −2.59565390105041595052110660134, −2.59004354622948420244506161721, −2.35821044369960577354605247759, −2.23076976950991820046287219988, −2.15330192963795843913373422387, −2.12527774224179111093683895933, −2.08501646360362318764886522573, −1.85379443832162937236185699105, −1.78398681083938149565740415732, −1.75252778204371813879674385188, −1.45327481786846935970663607410, −1.43202350084029815001157095984, −1.37265526782097777890976050624, −1.33064800379332359694997552608, −0.991137069325056654341010841400, −0.985723670048946161206812940610, −0.983877278254931319822383704258, −0.02943870749834837530785946662,
0.02943870749834837530785946662, 0.983877278254931319822383704258, 0.985723670048946161206812940610, 0.991137069325056654341010841400, 1.33064800379332359694997552608, 1.37265526782097777890976050624, 1.43202350084029815001157095984, 1.45327481786846935970663607410, 1.75252778204371813879674385188, 1.78398681083938149565740415732, 1.85379443832162937236185699105, 2.08501646360362318764886522573, 2.12527774224179111093683895933, 2.15330192963795843913373422387, 2.23076976950991820046287219988, 2.35821044369960577354605247759, 2.59004354622948420244506161721, 2.59565390105041595052110660134, 2.72279859696078191042970372981, 2.88361662340205635903268130157, 2.98790083069800652164428781202, 3.01902831265171959151218343778, 3.16415375258652697324958799728, 3.26294381228667612873956131774, 3.31245763798243748049238061128
Plot not available for L-functions of degree greater than 10.
