| L(s) = 1 | + 3·3-s − 6·11-s − 6·17-s + 6·19-s + 3·23-s − 12·27-s + 12·29-s − 3·31-s − 18·33-s − 6·37-s + 3·41-s − 9·43-s + 18·47-s − 18·51-s + 6·53-s + 18·57-s − 3·59-s + 9·61-s + 12·67-s + 9·69-s + 9·71-s − 12·73-s − 3·79-s − 18·81-s + 36·87-s + 6·89-s − 9·93-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.80·11-s − 1.45·17-s + 1.37·19-s + 0.625·23-s − 2.30·27-s + 2.22·29-s − 0.538·31-s − 3.13·33-s − 0.986·37-s + 0.468·41-s − 1.37·43-s + 2.62·47-s − 2.52·51-s + 0.824·53-s + 2.38·57-s − 0.390·59-s + 1.15·61-s + 1.46·67-s + 1.08·69-s + 1.06·71-s − 1.40·73-s − 0.337·79-s − 2·81-s + 3.85·87-s + 0.635·89-s − 0.933·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.274153462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.274153462\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 5 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 113 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 + 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 185 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 211 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 155 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 126 T^{2} - 715 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 167 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 84 T^{2} + 393 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 189 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 595 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 18 T + 228 T^{2} - 1799 T^{3} + 228 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T - 12 T^{2} + 441 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 153 T^{2} + 301 T^{3} + 153 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 1135 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 1024 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 132 T^{2} - 765 T^{3} + 132 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1560 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 471 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 192 T^{2} - 163 T^{3} + 192 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 369 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 21 T + 357 T^{2} + 3607 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.91414066254040351260500308275, −6.37702340421749590967600413718, −6.36593317032140093219071374424, −6.33336489406684406825413109084, −5.67145292156832161900205219351, −5.54109122439305824859504646644, −5.37088733249355273377345641895, −5.24645848441735856054898800782, −4.92239747856337510302527991798, −4.74385088340120640313128875920, −4.28681075647765287827531930184, −4.12202885478196821170847836396, −4.01513969921099836860815088631, −3.30380290991678808184116635229, −3.29793224270338873185346638814, −3.26253812351603968248419859406, −2.68501371170429341434972835815, −2.64557910832406592505553867166, −2.64292094840534728812399859398, −2.00313945969853743329480082789, −1.95798389095427717293791782209, −1.70538679090166193674771108843, −0.849420885407169467373512811395, −0.66716216194583303999643075606, −0.44076388390519203488949630887,
0.44076388390519203488949630887, 0.66716216194583303999643075606, 0.849420885407169467373512811395, 1.70538679090166193674771108843, 1.95798389095427717293791782209, 2.00313945969853743329480082789, 2.64292094840534728812399859398, 2.64557910832406592505553867166, 2.68501371170429341434972835815, 3.26253812351603968248419859406, 3.29793224270338873185346638814, 3.30380290991678808184116635229, 4.01513969921099836860815088631, 4.12202885478196821170847836396, 4.28681075647765287827531930184, 4.74385088340120640313128875920, 4.92239747856337510302527991798, 5.24645848441735856054898800782, 5.37088733249355273377345641895, 5.54109122439305824859504646644, 5.67145292156832161900205219351, 6.33336489406684406825413109084, 6.36593317032140093219071374424, 6.37702340421749590967600413718, 6.91414066254040351260500308275
