| L(s) = 1 | + 3·2-s − 4·3-s + 6·4-s − 7·5-s − 12·6-s − 9·7-s + 10·8-s + 4·9-s − 21·10-s − 13·11-s − 24·12-s − 13·13-s − 27·14-s + 28·15-s + 15·16-s − 8·17-s + 12·18-s − 5·19-s − 42·20-s + 36·21-s − 39·22-s − 9·23-s − 40·24-s + 20·25-s − 39·26-s + 9·27-s − 54·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 2.30·3-s + 3·4-s − 3.13·5-s − 4.89·6-s − 3.40·7-s + 3.53·8-s + 4/3·9-s − 6.64·10-s − 3.91·11-s − 6.92·12-s − 3.60·13-s − 7.21·14-s + 7.22·15-s + 15/4·16-s − 1.94·17-s + 2.82·18-s − 1.14·19-s − 9.39·20-s + 7.85·21-s − 8.31·22-s − 1.87·23-s − 8.16·24-s + 4·25-s − 7.64·26-s + 1.73·27-s − 10.2·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3011^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3011 | | \( 1+O(T) \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 23 T^{3} + 4 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 7 T + 29 T^{2} + 77 T^{3} + 29 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 9 T + 41 T^{2} + 125 T^{3} + 41 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 13 T + 87 T^{2} + 357 T^{3} + 87 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + p T + 93 T^{2} + 409 T^{3} + 93 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 8 T + 70 T^{2} + 285 T^{3} + 70 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 5 T + 21 T^{2} + 23 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 413 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 22 T + 8 p T^{2} + 1527 T^{3} + 8 p^{2} T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 2 T + 50 T^{2} - 3 T^{3} + 50 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 14 T + 167 T^{2} + 1092 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + 37 T^{2} - 255 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 38 T^{2} - 203 T^{3} + 38 p T^{4} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 98 T^{2} + 61 T^{3} + 98 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 4 T + 92 T^{2} - 395 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 7 T + 163 T^{2} - 819 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 14 T + 218 T^{2} + 1701 T^{3} + 218 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 192 T^{2} + 1659 T^{3} + 192 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 129 T^{2} + 987 T^{3} + 129 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 2 T + 120 T^{2} - 11 T^{3} + 120 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 20 T + 284 T^{2} + 2601 T^{3} + 284 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 137 T^{2} + 448 T^{3} + 137 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + T + 237 T^{2} + 135 T^{3} + 237 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + T + 205 T^{2} - 143 T^{3} + 205 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.36238063863473215685024513143, −7.19491280229722851032481720726, −7.17955607849783625532295179694, −6.87108820711518505958738636299, −6.55980525995023381034083768659, −6.44523647534896554272948287445, −5.88297858818533028071252505148, −5.83866748283632855031560313658, −5.69971814306452640445926012029, −5.39084932916360750117827407504, −5.31261084591392816594720188978, −4.92460973584157887601387916627, −4.83031749984607127365507291763, −4.42310420620210330453368974721, −4.26757194460697899683795734245, −3.96440069559322117728101305879, −3.79190523400630277505227095009, −3.44806509650657136136480829565, −3.16184399308235782601890804688, −2.88113922792060225793490564596, −2.75803529337297676465942988789, −2.32213880370160825301068173527, −2.25413871480268954243304941978, −1.90387730588182140982630630979, 0, 0, 0, 0, 0, 0,
1.90387730588182140982630630979, 2.25413871480268954243304941978, 2.32213880370160825301068173527, 2.75803529337297676465942988789, 2.88113922792060225793490564596, 3.16184399308235782601890804688, 3.44806509650657136136480829565, 3.79190523400630277505227095009, 3.96440069559322117728101305879, 4.26757194460697899683795734245, 4.42310420620210330453368974721, 4.83031749984607127365507291763, 4.92460973584157887601387916627, 5.31261084591392816594720188978, 5.39084932916360750117827407504, 5.69971814306452640445926012029, 5.83866748283632855031560313658, 5.88297858818533028071252505148, 6.44523647534896554272948287445, 6.55980525995023381034083768659, 6.87108820711518505958738636299, 7.17955607849783625532295179694, 7.19491280229722851032481720726, 7.36238063863473215685024513143
