| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 6·6-s + 3·7-s − 4·8-s + 6·9-s + 9·12-s + 5·13-s − 6·14-s + 3·16-s − 4·17-s − 12·18-s + 19-s + 9·21-s − 12·24-s − 10·26-s + 10·27-s + 9·28-s − 2·29-s + 17·31-s − 6·32-s + 8·34-s + 18·36-s − 2·38-s + 15·39-s − 4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 2.44·6-s + 1.13·7-s − 1.41·8-s + 2·9-s + 2.59·12-s + 1.38·13-s − 1.60·14-s + 3/4·16-s − 0.970·17-s − 2.82·18-s + 0.229·19-s + 1.96·21-s − 2.44·24-s − 1.96·26-s + 1.92·27-s + 1.70·28-s − 0.371·29-s + 3.05·31-s − 1.06·32-s + 1.37·34-s + 3·36-s − 0.324·38-s + 2.40·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.289626136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.289626136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 2 p T^{2} - 25 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 122 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 26 T^{2} + 158 T^{3} + 26 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 63 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 26 T^{2} + 58 T^{3} + 26 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} + 132 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 34 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 122 T^{2} + 326 T^{3} + 122 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 17 T + 97 T^{2} - 362 T^{3} + 97 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 166 T^{2} - 948 T^{3} + 166 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} + 188 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 146 T^{2} + 572 T^{3} + 146 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 95 T^{2} - 10 p T^{3} + 95 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 7 T + 9 T^{2} - 650 T^{3} + 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 430 T^{2} - 4272 T^{3} + 430 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 175 T^{2} + 988 T^{3} + 175 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 200 T^{2} + 736 T^{3} + 200 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 33 T^{2} + 4 p T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 167 T^{2} - 28 T^{3} + 167 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 29 T + 366 T^{2} + 3473 T^{3} + 366 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.05809799011028904744750972618, −6.67280301380285580738227548119, −6.49614565028758058902556663634, −6.16177331898411215237106747813, −6.13421413649613026653464758475, −5.86396186061814172854583397782, −5.44812547212522015382316516323, −5.15445332491733764757124599151, −4.90253866841712723387862683318, −4.66152439537371103892303874111, −4.32940804431582984408361640379, −4.18545952845182959659214357810, −3.89757509607740035571604150677, −3.71092064854082877092638272587, −3.32020472940133227361624131541, −3.09688086523261067651874353973, −2.72956488628604900771944456427, −2.48752638541603907398817592230, −2.31536321485713055993466343543, −2.12613638385070312616862171116, −1.68644685431640718539587238266, −1.41376748294642212951909696495, −1.11156754632507947773457392790, −0.863922461057985756973145106078, −0.42703294550520247560253179368,
0.42703294550520247560253179368, 0.863922461057985756973145106078, 1.11156754632507947773457392790, 1.41376748294642212951909696495, 1.68644685431640718539587238266, 2.12613638385070312616862171116, 2.31536321485713055993466343543, 2.48752638541603907398817592230, 2.72956488628604900771944456427, 3.09688086523261067651874353973, 3.32020472940133227361624131541, 3.71092064854082877092638272587, 3.89757509607740035571604150677, 4.18545952845182959659214357810, 4.32940804431582984408361640379, 4.66152439537371103892303874111, 4.90253866841712723387862683318, 5.15445332491733764757124599151, 5.44812547212522015382316516323, 5.86396186061814172854583397782, 6.13421413649613026653464758475, 6.16177331898411215237106747813, 6.49614565028758058902556663634, 6.67280301380285580738227548119, 7.05809799011028904744750972618
