| L(s) = 1 | − 2-s + 3·3-s − 4-s − 3·6-s − 2·7-s + 8-s + 6·9-s − 4·11-s − 3·12-s − 8·13-s + 2·14-s − 16-s − 2·17-s − 6·18-s + 2·19-s − 6·21-s + 4·22-s + 10·23-s + 3·24-s + 8·26-s + 10·27-s + 2·28-s − 6·29-s − 2·31-s + 32-s − 12·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s − 1.20·11-s − 0.866·12-s − 2.21·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 1.41·18-s + 0.458·19-s − 1.30·21-s + 0.852·22-s + 2.08·23-s + 0.612·24-s + 1.56·26-s + 1.92·27-s + 0.377·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s − 2.08·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 204 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 28 T^{2} + 6 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 68 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 95 T^{2} - 476 T^{3} + 95 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 262 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 2 T^{2} - 132 T^{3} + 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 228 T^{2} + 1646 T^{3} + 228 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 10 T^{2} - 296 T^{3} + 10 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 106 T^{2} + 384 T^{3} + 106 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1452 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 137 T^{2} + 976 T^{3} + 137 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 188 T^{2} + 930 T^{3} + 188 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 32 T + 550 T^{2} + 5712 T^{3} + 550 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 120 T^{2} + 130 T^{3} + 120 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 305 T^{2} + 3192 T^{3} + 305 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 259 T^{2} - 2028 T^{3} + 259 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 305 T^{2} - 2180 T^{3} + 305 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.350600399829644404213767338259, −7.77617116319020980721399760843, −7.56933336112154433308458745834, −7.35714497176918421924412916156, −7.23378233155660421377559862960, −7.14681331188341994201030008472, −6.80327244403889353506563749747, −6.48274839844457378445551715362, −5.97545584334663221113782949440, −5.95084337184628754861797241834, −5.29200710090638498718022769336, −5.09269822617481137787959662868, −5.03686632399848423487136565687, −4.63594507790781220186349672185, −4.46288489630853776979665172855, −4.26218515015850950467418644485, −3.46407360391253738855499616289, −3.44453100170858720252082631230, −3.07200470383600315643773752296, −2.98412572763016617984641321195, −2.79500784751257801772679239756, −2.14920749114021160937024689054, −2.00599056235957502266534841027, −1.51338258319463742068301335839, −1.31674261327034211826305082261, 0, 0, 0,
1.31674261327034211826305082261, 1.51338258319463742068301335839, 2.00599056235957502266534841027, 2.14920749114021160937024689054, 2.79500784751257801772679239756, 2.98412572763016617984641321195, 3.07200470383600315643773752296, 3.44453100170858720252082631230, 3.46407360391253738855499616289, 4.26218515015850950467418644485, 4.46288489630853776979665172855, 4.63594507790781220186349672185, 5.03686632399848423487136565687, 5.09269822617481137787959662868, 5.29200710090638498718022769336, 5.95084337184628754861797241834, 5.97545584334663221113782949440, 6.48274839844457378445551715362, 6.80327244403889353506563749747, 7.14681331188341994201030008472, 7.23378233155660421377559862960, 7.35714497176918421924412916156, 7.56933336112154433308458745834, 7.77617116319020980721399760843, 8.350600399829644404213767338259
