| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s + 5-s − 9·6-s − 7-s + 10·8-s − 3·9-s + 3·10-s + 9·11-s − 18·12-s − 4·13-s − 3·14-s − 3·15-s + 15·16-s − 7·17-s − 9·18-s + 2·19-s + 6·20-s + 3·21-s + 27·22-s − 8·23-s − 30·24-s − 10·25-s − 12·26-s + 26·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 0.447·5-s − 3.67·6-s − 0.377·7-s + 3.53·8-s − 9-s + 0.948·10-s + 2.71·11-s − 5.19·12-s − 1.10·13-s − 0.801·14-s − 0.774·15-s + 15/4·16-s − 1.69·17-s − 2.12·18-s + 0.458·19-s + 1.34·20-s + 0.654·21-s + 5.75·22-s − 1.66·23-s − 6.12·24-s − 2·25-s − 2.35·26-s + 5.00·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 2011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 2011^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 2011 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 5 | $A_4\times C_2$ | \( 1 - T + 11 T^{2} - 11 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + T + 17 T^{2} + 15 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 9 T + 47 T^{2} - 173 T^{3} + 47 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 4 T + 27 T^{2} + 64 T^{3} + 27 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 37 T^{2} + 165 T^{3} + 37 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 2 T + 54 T^{2} - 71 T^{3} + 54 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 86 T^{2} + 373 T^{3} + 86 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 20 T + 216 T^{2} + 1425 T^{3} + 216 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 19 T^{2} - 85 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 9 T + 125 T^{2} + 641 T^{3} + 125 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 15 T^{2} + 255 T^{3} + 15 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 11 T + 165 T^{2} + 977 T^{3} + 165 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 16 T + 105 T^{2} - 504 T^{3} + 105 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 6 T + 158 T^{2} + 605 T^{3} + 158 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 153 T^{3} + 22 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 144 T^{2} + 65 T^{3} + 144 p T^{4} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 2 T + 16 T^{2} - 497 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 8 T + 217 T^{2} + 1128 T^{3} + 217 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 8 T + 158 T^{2} + 705 T^{3} + 158 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 25 T + 337 T^{2} - 3325 T^{3} + 337 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 105 T^{2} + 387 T^{3} + 105 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 25 T + 367 T^{2} + 3825 T^{3} + 367 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 20 T + 342 T^{2} - 3365 T^{3} + 342 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81095704755795835359319952335, −7.44005559551728156453836012056, −6.96777032702007001899118250526, −6.74225695801388547650352843346, −6.68423622058871550570606709741, −6.62309208920454445400055414969, −6.15226397900925147025580893112, −5.93653757979845474123413662323, −5.81486099072706034374290631077, −5.73272805835513430610896125516, −5.22497831121878040808309733017, −5.17868827713042084331422082530, −5.17472820698822119621048710660, −4.43596262246103765783008302457, −4.27402493162757434036164813006, −4.14866734392034142677055079622, −3.61827744899941120707033427676, −3.53401012009422937020358616168, −3.51994380997967697138954762978, −2.82462464203504724982360661771, −2.52403188736841675551328978437, −2.22704356991432371312954094578, −1.87276296044193285885650575282, −1.56351271849862850327584629778, −1.36402112361311783527842405851, 0, 0, 0,
1.36402112361311783527842405851, 1.56351271849862850327584629778, 1.87276296044193285885650575282, 2.22704356991432371312954094578, 2.52403188736841675551328978437, 2.82462464203504724982360661771, 3.51994380997967697138954762978, 3.53401012009422937020358616168, 3.61827744899941120707033427676, 4.14866734392034142677055079622, 4.27402493162757434036164813006, 4.43596262246103765783008302457, 5.17472820698822119621048710660, 5.17868827713042084331422082530, 5.22497831121878040808309733017, 5.73272805835513430610896125516, 5.81486099072706034374290631077, 5.93653757979845474123413662323, 6.15226397900925147025580893112, 6.62309208920454445400055414969, 6.68423622058871550570606709741, 6.74225695801388547650352843346, 6.96777032702007001899118250526, 7.44005559551728156453836012056, 7.81095704755795835359319952335
