| L(s) = 1 | − 3·3-s − 5-s + 3·7-s + 9-s − 6·11-s − 4·13-s + 3·15-s − 3·17-s − 2·19-s − 9·21-s + 6·23-s − 9·25-s + 7·27-s + 4·29-s + 13·31-s + 18·33-s − 3·35-s + 12·39-s − 5·41-s − 5·43-s − 45-s + 6·47-s + 6·49-s + 9·51-s + 9·53-s + 6·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.774·15-s − 0.727·17-s − 0.458·19-s − 1.96·21-s + 1.25·23-s − 9/5·25-s + 1.34·27-s + 0.742·29-s + 2.33·31-s + 3.13·33-s − 0.507·35-s + 1.92·39-s − 0.780·41-s − 0.762·43-s − 0.149·45-s + 0.875·47-s + 6/7·49-s + 1.26·51-s + 1.23·53-s + 0.809·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 14 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 12 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $D_{6}$ | \( 1 + 4 T + 19 T^{2} + 88 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 44 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 65 T^{2} - 244 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 264 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 13 T + 134 T^{2} - 798 T^{3} + 134 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 47 T^{2} + 64 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 54 T^{2} + 24 T^{3} + 54 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 98 T^{2} + 282 T^{3} + 98 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 532 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 104 T^{2} - 628 T^{3} + 104 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 217 T^{2} + 1620 T^{3} + 217 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 9 T + 206 T^{2} + 1112 T^{3} + 206 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 120 T^{2} + 1150 T^{3} + 120 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 24 T + 341 T^{2} - 3472 T^{3} + 341 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 78 T^{2} + 304 T^{3} + 78 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 452 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 65 T^{2} - 424 T^{3} + 65 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 227 T^{2} - 1096 T^{3} + 227 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 58 T^{2} + 364 T^{3} + 58 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.34232638594565110909754729723, −6.91237958750604997656684993434, −6.79573202830111803656852247913, −6.69683219646470153877371005319, −6.26143603383762404615292062713, −6.12919939572968740495190648217, −5.86431424178231375638596966912, −5.50304933992580213337637674143, −5.44347371595273215444274781698, −5.29403284134411650899246973738, −4.85700656403506434248818030698, −4.81435933822960201284560913261, −4.61722837516255419537215337869, −4.40038609909755662978463709342, −4.08483422244643669341178264896, −3.82610190251735905029581365947, −3.22866357463142974798718958764, −3.03113622062427515629637918844, −2.93320902731052017577362319596, −2.34493407984308567028685218068, −2.30191459103532578766357660923, −2.13076633927988678615073528120, −1.52104452918059723754430541003, −0.990851500381288648914857692131, −0.932711826185541892155116411525, 0, 0, 0,
0.932711826185541892155116411525, 0.990851500381288648914857692131, 1.52104452918059723754430541003, 2.13076633927988678615073528120, 2.30191459103532578766357660923, 2.34493407984308567028685218068, 2.93320902731052017577362319596, 3.03113622062427515629637918844, 3.22866357463142974798718958764, 3.82610190251735905029581365947, 4.08483422244643669341178264896, 4.40038609909755662978463709342, 4.61722837516255419537215337869, 4.81435933822960201284560913261, 4.85700656403506434248818030698, 5.29403284134411650899246973738, 5.44347371595273215444274781698, 5.50304933992580213337637674143, 5.86431424178231375638596966912, 6.12919939572968740495190648217, 6.26143603383762404615292062713, 6.69683219646470153877371005319, 6.79573202830111803656852247913, 6.91237958750604997656684993434, 7.34232638594565110909754729723
