| L(s) = 1 | − 3·3-s − 2·4-s + 2·5-s − 3·7-s + 8-s + 6·9-s − 4·11-s + 6·12-s + 10·13-s − 6·15-s − 7·17-s + 14·19-s − 4·20-s + 9·21-s − 9·23-s − 3·24-s − 7·25-s − 10·27-s + 6·28-s + 9·29-s − 8·31-s − 4·32-s + 12·33-s − 6·35-s − 12·36-s − 37-s − 30·39-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s + 0.894·5-s − 1.13·7-s + 0.353·8-s + 2·9-s − 1.20·11-s + 1.73·12-s + 2.77·13-s − 1.54·15-s − 1.69·17-s + 3.21·19-s − 0.894·20-s + 1.96·21-s − 1.87·23-s − 0.612·24-s − 7/5·25-s − 1.92·27-s + 1.13·28-s + 1.67·29-s − 1.43·31-s − 0.707·32-s + 2.08·33-s − 1.01·35-s − 2·36-s − 0.164·37-s − 4.80·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 191^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 191^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552770108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552770108\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 191 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} - T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 11 T^{2} - 13 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 32 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 10 T + 66 T^{2} - 274 T^{3} + 66 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + p T^{2} + 2 T^{3} + p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 14 T + 116 T^{2} - 606 T^{3} + 116 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 80 T^{2} + 385 T^{3} + 80 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 82 T^{2} + 482 T^{3} + 82 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + T + 84 T^{2} + 10 T^{3} + 84 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 156 T^{2} - 1152 T^{3} + 156 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1040 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 67 T^{2} + 342 T^{3} + 67 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 59 T^{2} - 125 T^{3} + 59 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + T + 100 T^{2} + 6 p T^{3} + 100 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 325 T^{2} - 2911 T^{3} + 325 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 17 T + 292 T^{2} + 2426 T^{3} + 292 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 113 T^{2} + 340 T^{3} + 113 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 26 T + 417 T^{2} - 4265 T^{3} + 417 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 310 T^{2} - 2602 T^{3} + 310 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 19 T + 319 T^{2} + 2958 T^{3} + 319 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 7 T + 226 T^{2} - 1038 T^{3} + 226 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 7 T + 6 T^{2} - 1244 T^{3} + 6 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.45304508478930949409125701634, −7.30855755687492464258846838049, −6.85785141330466255198531989748, −6.53295085104707978600229004340, −6.29717176706829382061657800519, −6.28249768223238983459308463717, −5.91497380326018169811233512744, −5.69128864789993793877536935360, −5.51337100763001135609865203259, −5.44036290055427168758799482411, −5.01704181867724935719163898209, −4.82429379188155609990241812197, −4.38505883248875814239641039751, −4.04899434164127530734691093792, −3.88899786453654772652973050030, −3.87244444709663126090545606047, −3.33567700651976668361911816105, −3.07191739654610702155541478533, −2.49199329038183555160502862064, −2.36341730515409573553164553524, −1.75441985654495871116973962895, −1.59274361085997758472205017177, −1.01718915308433137445119930067, −0.55881691903650565761123273360, −0.49059418597479321147403389026,
0.49059418597479321147403389026, 0.55881691903650565761123273360, 1.01718915308433137445119930067, 1.59274361085997758472205017177, 1.75441985654495871116973962895, 2.36341730515409573553164553524, 2.49199329038183555160502862064, 3.07191739654610702155541478533, 3.33567700651976668361911816105, 3.87244444709663126090545606047, 3.88899786453654772652973050030, 4.04899434164127530734691093792, 4.38505883248875814239641039751, 4.82429379188155609990241812197, 5.01704181867724935719163898209, 5.44036290055427168758799482411, 5.51337100763001135609865203259, 5.69128864789993793877536935360, 5.91497380326018169811233512744, 6.28249768223238983459308463717, 6.29717176706829382061657800519, 6.53295085104707978600229004340, 6.85785141330466255198531989748, 7.30855755687492464258846838049, 7.45304508478930949409125701634
