| L(s) = 1 | − 9·3-s + 4·5-s − 30·7-s + 54·9-s + 16·11-s + 39·13-s − 36·15-s − 146·17-s − 94·19-s + 270·21-s + 48·23-s − 107·25-s − 270·27-s − 2·29-s − 302·31-s − 144·33-s − 120·35-s + 374·37-s − 351·39-s + 480·41-s + 260·43-s + 216·45-s + 24·47-s + 159·49-s + 1.31e3·51-s − 678·53-s + 64·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.357·5-s − 1.61·7-s + 2·9-s + 0.438·11-s + 0.832·13-s − 0.619·15-s − 2.08·17-s − 1.13·19-s + 2.80·21-s + 0.435·23-s − 0.855·25-s − 1.92·27-s − 0.0128·29-s − 1.74·31-s − 0.759·33-s − 0.579·35-s + 1.66·37-s − 1.44·39-s + 1.82·41-s + 0.922·43-s + 0.715·45-s + 0.0744·47-s + 0.463·49-s + 3.60·51-s − 1.75·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.255796322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255796322\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 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
−9.225905207323683479205533910662, −8.821579661938653607133109979750, −8.524297692716032559794292086279, −8.351202003037429095931300792865, −7.57668720978151822466820013357, −7.38411910005463762053578377412, −7.24287610587851819813154069337, −6.71486153609487080740571710866, −6.38997823609475309454584224457, −6.34277758221271555312520944227, −6.01378221147016721218867316155, −5.89337886571642910530731103798, −5.54621668960216646944113105638, −4.84813685901376775360790668282, −4.61169678387003034692640910242, −4.53299357971379930481855150297, −3.76977568213325285150778648295, −3.59326293987790616391690387734, −3.53651538350112208893851713361, −2.41046654086293974831104480994, −2.22477015589353256171463346938, −2.01748080276262735111020109015, −1.07046109315318737750730796684, −0.59978760653450503243651524009, −0.39328445178208225741188991741,
0.39328445178208225741188991741, 0.59978760653450503243651524009, 1.07046109315318737750730796684, 2.01748080276262735111020109015, 2.22477015589353256171463346938, 2.41046654086293974831104480994, 3.53651538350112208893851713361, 3.59326293987790616391690387734, 3.76977568213325285150778648295, 4.53299357971379930481855150297, 4.61169678387003034692640910242, 4.84813685901376775360790668282, 5.54621668960216646944113105638, 5.89337886571642910530731103798, 6.01378221147016721218867316155, 6.34277758221271555312520944227, 6.38997823609475309454584224457, 6.71486153609487080740571710866, 7.24287610587851819813154069337, 7.38411910005463762053578377412, 7.57668720978151822466820013357, 8.351202003037429095931300792865, 8.524297692716032559794292086279, 8.821579661938653607133109979750, 9.225905207323683479205533910662
