| L(s) = 1 | + 3·2-s + 3·4-s + 3·7-s − 3·11-s + 3·13-s + 9·14-s − 3·16-s + 9·17-s − 3·19-s − 9·22-s + 6·23-s + 9·26-s + 9·28-s − 12·29-s − 12·31-s − 6·32-s + 27·34-s + 3·37-s − 9·38-s + 3·41-s + 12·43-s − 9·44-s + 18·46-s − 6·47-s − 6·49-s + 9·52-s + 18·53-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 1.13·7-s − 0.904·11-s + 0.832·13-s + 2.40·14-s − 3/4·16-s + 2.18·17-s − 0.688·19-s − 1.91·22-s + 1.25·23-s + 1.76·26-s + 1.70·28-s − 2.22·29-s − 2.15·31-s − 1.06·32-s + 4.63·34-s + 0.493·37-s − 1.45·38-s + 0.468·41-s + 1.82·43-s − 1.35·44-s + 2.65·46-s − 0.875·47-s − 6/7·49-s + 1.24·52-s + 2.47·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.59184506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.59184506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 63 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 61 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 639 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 763 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 87 T^{2} - 223 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 27 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 297 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 240 T^{2} - 1989 T^{3} + 240 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 21 T + 321 T^{2} - 2799 T^{3} + 321 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 695 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 279 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 479 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 186 T^{2} - 1001 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 6 T + 222 T^{2} + 945 T^{3} + 222 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} - 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 222 T^{2} + 2891 T^{3} + 222 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.08927211780181096023532058359, −7.00344915286461969251024106589, −6.53152243492050434574558962285, −6.37545561679031064017049929100, −5.79000939266282429127586225522, −5.62776032283456124287936366983, −5.57146046638486826574026315769, −5.32664882251355969332092132545, −5.30706310476443586897020437051, −5.23659989248036097669727119642, −4.46693832214843610909082651515, −4.44008021993274159914466515105, −4.33737452671504580656740249597, −3.77625435613680924832916288952, −3.74860501883706702640869536010, −3.70410729153630710114023959693, −3.20319400381982646828516155318, −2.93612124537189257714551492910, −2.64580802490679142630004577841, −2.27111768109537915838583459630, −1.94507418025516862670667619824, −1.58726570522028227413484264095, −1.37293628271581631243672623373, −0.70361751561108712567944034427, −0.51502978779912567328906272143,
0.51502978779912567328906272143, 0.70361751561108712567944034427, 1.37293628271581631243672623373, 1.58726570522028227413484264095, 1.94507418025516862670667619824, 2.27111768109537915838583459630, 2.64580802490679142630004577841, 2.93612124537189257714551492910, 3.20319400381982646828516155318, 3.70410729153630710114023959693, 3.74860501883706702640869536010, 3.77625435613680924832916288952, 4.33737452671504580656740249597, 4.44008021993274159914466515105, 4.46693832214843610909082651515, 5.23659989248036097669727119642, 5.30706310476443586897020437051, 5.32664882251355969332092132545, 5.57146046638486826574026315769, 5.62776032283456124287936366983, 5.79000939266282429127586225522, 6.37545561679031064017049929100, 6.53152243492050434574558962285, 7.00344915286461969251024106589, 7.08927211780181096023532058359
