L(s) = 1 | + 3·3-s − 3·5-s + 6·9-s + 4·11-s + 6·13-s − 9·15-s + 2·17-s − 3·19-s + 4·23-s + 6·25-s + 10·27-s + 2·29-s + 8·31-s + 12·33-s + 6·37-s + 18·39-s − 2·41-s + 12·43-s − 18·45-s − 4·47-s − 7·49-s + 6·51-s + 2·53-s − 12·55-s − 9·57-s + 12·59-s + 14·61-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.34·5-s + 2·9-s + 1.20·11-s + 1.66·13-s − 2.32·15-s + 0.485·17-s − 0.688·19-s + 0.834·23-s + 6/5·25-s + 1.92·27-s + 0.371·29-s + 1.43·31-s + 2.08·33-s + 0.986·37-s + 2.88·39-s − 0.312·41-s + 1.82·43-s − 2.68·45-s − 0.583·47-s − 49-s + 0.840·51-s + 0.274·53-s − 1.61·55-s − 1.19·57-s + 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.858276672\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.858276672\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 7 | $S_4\times C_2$ | \( 1 + p T^{2} + 16 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 7 T^{2} + 8 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 6 T + 37 T^{2} - 120 T^{3} + 37 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 52 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 152 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} + 40 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 448 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 109 T^{2} - 408 T^{3} + 109 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 163 T^{2} - 1024 T^{3} + 163 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 121 T^{2} + 344 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 135 T^{2} - 196 T^{3} + 135 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 169 T^{2} - 1128 T^{3} + 169 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 151 T^{2} - 1132 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 105 T^{2} - 408 T^{3} + 105 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 368 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 141 T^{2} + 504 T^{3} + 141 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 35 T^{2} + 1464 T^{3} - 35 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 141 T^{2} - 152 T^{3} + 141 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 30 T + 577 T^{2} - 6664 T^{3} + 577 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.009212005777289615762675737614, −7.940183865920907529661633920007, −7.58170819321746709761394199676, −7.36933996038771953185415902185, −6.81368976799422534631332608031, −6.80659411710702453354003421104, −6.66051010929653275573036045635, −6.23994864773986528664343334069, −5.96190993093039134516650147080, −5.79337705687277506358738772485, −5.00867922508602979870863409080, −4.87037418214582384459794863242, −4.80882879160165385228281569500, −4.10861342803061305614267072013, −4.08056402977940687716719507696, −3.86747730155700611493577351309, −3.38004494196964115295074554930, −3.37610598614071760537082366105, −3.18714946018868992964988309019, −2.36299867930553443406667395318, −2.31341029416785165323438360657, −2.05575474694119080414560541931, −1.13635636947267084829378209320, −0.952065895119842998882988728539, −0.833110127489546082984787077796,
0.833110127489546082984787077796, 0.952065895119842998882988728539, 1.13635636947267084829378209320, 2.05575474694119080414560541931, 2.31341029416785165323438360657, 2.36299867930553443406667395318, 3.18714946018868992964988309019, 3.37610598614071760537082366105, 3.38004494196964115295074554930, 3.86747730155700611493577351309, 4.08056402977940687716719507696, 4.10861342803061305614267072013, 4.80882879160165385228281569500, 4.87037418214582384459794863242, 5.00867922508602979870863409080, 5.79337705687277506358738772485, 5.96190993093039134516650147080, 6.23994864773986528664343334069, 6.66051010929653275573036045635, 6.80659411710702453354003421104, 6.81368976799422534631332608031, 7.36933996038771953185415902185, 7.58170819321746709761394199676, 7.940183865920907529661633920007, 8.009212005777289615762675737614