| L(s) = 1 | − 2·2-s + 9·3-s − 5·4-s − 4·5-s − 18·6-s − 30·7-s + 20·8-s + 54·9-s + 8·10-s + 16·11-s − 45·12-s + 60·14-s − 36·15-s − 51·16-s − 146·17-s − 108·18-s − 94·19-s + 20·20-s − 270·21-s − 32·22-s − 48·23-s + 180·24-s − 107·25-s + 270·27-s + 150·28-s − 2·29-s + 72·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 5/8·4-s − 0.357·5-s − 1.22·6-s − 1.61·7-s + 0.883·8-s + 2·9-s + 0.252·10-s + 0.438·11-s − 1.08·12-s + 1.14·14-s − 0.619·15-s − 0.796·16-s − 2.08·17-s − 1.41·18-s − 1.13·19-s + 0.223·20-s − 2.80·21-s − 0.310·22-s − 0.435·23-s + 1.53·24-s − 0.855·25-s + 1.92·27-s + 1.01·28-s − 0.0128·29-s + 0.438·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + 9 T^{2} + p^{3} T^{3} + 9 p^{3} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 123 T^{2} + 1864 T^{3} + 123 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 30 T + 741 T^{2} + 18596 T^{3} + 741 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 16 T + 1737 T^{2} - 72928 T^{3} + 1737 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 146 T + 20799 T^{2} + 1505852 T^{3} + 20799 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 94 T + 6145 T^{2} + 509876 T^{3} + 6145 p^{3} T^{4} + 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 48 T + 15573 T^{2} + 1702560 T^{3} + 15573 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 63051 T^{2} - 101620 T^{3} + 63051 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 302 T + 71837 T^{2} + 10796516 T^{3} + 71837 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 374 T + 114995 T^{2} + 30130340 T^{3} + 114995 p^{3} T^{4} + 374 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 480 T + 208479 T^{2} + 53244336 T^{3} + 208479 p^{3} T^{4} + 480 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 260 T + 200425 T^{2} + 37680472 T^{3} + 200425 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 24 T + 142989 T^{2} - 23086032 T^{3} + 142989 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 678 T + 404403 T^{2} + 200405604 T^{3} + 404403 p^{3} T^{4} + 678 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1788 T + 1572249 T^{2} - 871859112 T^{3} + 1572249 p^{3} T^{4} - 1788 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 230 T + 636491 T^{2} - 98131748 T^{3} + 636491 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 74 T + 493073 T^{2} + 40252028 T^{3} + 493073 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 948 T + 1061157 T^{2} - 608134872 T^{3} + 1061157 p^{3} T^{4} - 948 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 222 T + 223815 T^{2} - 195504100 T^{3} + 223815 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 24 T + 1400781 T^{2} + 31423696 T^{3} + 1400781 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 796 T + 1904433 T^{2} - 924248872 T^{3} + 1904433 p^{3} T^{4} - 796 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1436 T + 2536191 T^{2} + 2054800856 T^{3} + 2536191 p^{3} T^{4} + 1436 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3242 T + 6203519 T^{2} + 7136252780 T^{3} + 6203519 p^{3} T^{4} + 3242 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.728765990291032714829583594327, −9.350780922537508244785913159765, −9.031964806703631303514623582214, −9.021358536086264161436550421648, −8.468291784352856746961559863739, −8.428771475042315446601100784615, −8.320420368105669451645292705645, −7.70479024080982071698394707677, −7.48252064690124811197209155735, −6.85711038441979435690793613026, −6.68761715862628168139875942250, −6.60790906582811672820474635235, −6.53498856713253695947799233301, −5.54805200134730677429346047917, −5.21662785226880525030809181435, −5.00990637960869075267089091006, −4.16300937100239495933003618680, −4.11034989743738209494486831321, −3.90245963438468504432039985416, −3.50610358684676637736631739014, −3.15542986430751243616492125625, −2.44575445381826246487359122719, −2.22989864459975986446525143615, −1.81679442176388257846237925670, −1.31101060148938842872909730105, 0, 0, 0,
1.31101060148938842872909730105, 1.81679442176388257846237925670, 2.22989864459975986446525143615, 2.44575445381826246487359122719, 3.15542986430751243616492125625, 3.50610358684676637736631739014, 3.90245963438468504432039985416, 4.11034989743738209494486831321, 4.16300937100239495933003618680, 5.00990637960869075267089091006, 5.21662785226880525030809181435, 5.54805200134730677429346047917, 6.53498856713253695947799233301, 6.60790906582811672820474635235, 6.68761715862628168139875942250, 6.85711038441979435690793613026, 7.48252064690124811197209155735, 7.70479024080982071698394707677, 8.320420368105669451645292705645, 8.428771475042315446601100784615, 8.468291784352856746961559863739, 9.021358536086264161436550421648, 9.031964806703631303514623582214, 9.350780922537508244785913159765, 9.728765990291032714829583594327
