| L(s) = 1 | + 3·3-s + 3·5-s + 2·7-s + 6·9-s + 2·11-s + 9·15-s − 6·17-s + 3·19-s + 6·21-s + 12·23-s + 6·25-s + 10·27-s + 4·29-s + 4·31-s + 6·33-s + 6·35-s − 8·37-s + 14·43-s + 18·45-s + 12·47-s − 3·49-s − 18·51-s + 6·53-s + 6·55-s + 9·57-s + 16·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.755·7-s + 2·9-s + 0.603·11-s + 2.32·15-s − 1.45·17-s + 0.688·19-s + 1.30·21-s + 2.50·23-s + 6/5·25-s + 1.92·27-s + 0.742·29-s + 0.718·31-s + 1.04·33-s + 1.01·35-s − 1.31·37-s + 2.13·43-s + 2.68·45-s + 1.75·47-s − 3/7·49-s − 2.52·51-s + 0.824·53-s + 0.809·55-s + 1.19·57-s + 2.08·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.20817931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.20817931\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 76 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 17 | $D_{6}$ | \( 1 + 6 T + 35 T^{2} + 140 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 12 T + 89 T^{2} - 488 T^{3} + 89 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 85 T^{2} - 228 T^{3} + 85 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} + 8 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 117 T^{2} + 548 T^{3} + 117 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 65 T^{2} + 124 T^{3} + 65 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 139 T^{2} - 908 T^{3} + 139 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 143 T^{2} - 572 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 16 T + 233 T^{2} - 1856 T^{3} + 233 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 39 T^{2} + 556 T^{3} + 39 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 125 T^{2} - 616 T^{3} + 125 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 151 T^{2} + 284 T^{3} + 151 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} + 24 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 81 T^{2} + 516 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 185 T^{2} + 900 T^{3} + 185 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.41924121037263209694401441584, −7.05323430166268022863876290340, −6.97661411185707920819498895304, −6.77621103687427947642378131460, −6.46947821770975578865892590576, −6.11913622713502240646875167457, −6.09673648229398328550240466067, −5.48611932564853712236000164268, −5.33946185409008379901174344745, −5.05972187290822889818492981446, −4.89613470246475703920396556132, −4.56447920738468185414852580008, −4.40479018708109083822113919629, −3.85076342794092243991717441785, −3.73646607515698203792198350918, −3.64451772136082626038309899625, −2.90736949869887144200165221752, −2.78236509784746885664175478420, −2.70171575211544791274023813309, −2.25073185862266782942922809410, −2.07185647403809080253698421841, −1.73957985921372683655496556478, −1.20998597371058726737610643989, −0.998472892298478562306170931813, −0.73685881756563129953925303130,
0.73685881756563129953925303130, 0.998472892298478562306170931813, 1.20998597371058726737610643989, 1.73957985921372683655496556478, 2.07185647403809080253698421841, 2.25073185862266782942922809410, 2.70171575211544791274023813309, 2.78236509784746885664175478420, 2.90736949869887144200165221752, 3.64451772136082626038309899625, 3.73646607515698203792198350918, 3.85076342794092243991717441785, 4.40479018708109083822113919629, 4.56447920738468185414852580008, 4.89613470246475703920396556132, 5.05972187290822889818492981446, 5.33946185409008379901174344745, 5.48611932564853712236000164268, 6.09673648229398328550240466067, 6.11913622713502240646875167457, 6.46947821770975578865892590576, 6.77621103687427947642378131460, 6.97661411185707920819498895304, 7.05323430166268022863876290340, 7.41924121037263209694401441584
