| L(s) = 1 | − 3·2-s − 3·3-s + 7·4-s − 6·5-s + 9·6-s − 2·7-s − 14·8-s + 6·9-s + 18·10-s − 5·11-s − 21·12-s + 6·14-s + 18·15-s + 21·16-s − 17-s − 18·18-s + 7·19-s − 42·20-s + 6·21-s + 15·22-s + 42·24-s + 16·25-s − 10·27-s − 14·28-s − 2·29-s − 54·30-s + 16·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 7/2·4-s − 2.68·5-s + 3.67·6-s − 0.755·7-s − 4.94·8-s + 2·9-s + 5.69·10-s − 1.50·11-s − 6.06·12-s + 1.60·14-s + 4.64·15-s + 21/4·16-s − 0.242·17-s − 4.24·18-s + 1.60·19-s − 9.39·20-s + 1.30·21-s + 3.19·22-s + 8.57·24-s + 16/5·25-s − 1.92·27-s − 2.64·28-s − 0.371·29-s − 9.85·30-s + 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 + 3 T + p T^{2} - T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 47 T^{3} + 4 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 T + 20 T^{2} + 27 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 25 T^{2} + 69 T^{3} + 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 7 T + 71 T^{2} - 273 T^{3} + 71 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} + 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 16 T + 134 T^{2} - 795 T^{3} + 134 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 22 T + 270 T^{2} + 2005 T^{3} + 270 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 11 T + 147 T^{2} + 873 T^{3} + 147 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 15 T + 176 T^{2} + 1331 T^{3} + 176 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 7 T + 155 T^{2} + 665 T^{3} + 155 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 161 T^{2} - 604 T^{3} + 161 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 11 T + 155 T^{2} + 1515 T^{3} + 155 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 122 T^{2} - 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 84 T^{2} - 47 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 290 T^{2} + 2035 T^{3} + 290 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + T + 167 T^{2} + 65 T^{3} + 167 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 5 T + 10 T^{2} + 667 T^{3} + 10 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.24632024213934150814357778698, −9.882691801179596796962295591839, −9.709876464886236275882495479204, −9.571202944388799864132972588980, −8.758256292235516698379198558779, −8.546423511461009828464401955907, −8.424268292688031193921047813833, −7.955365973166934187778379192542, −7.86195856353724085969307969484, −7.41768819056655327024411483281, −7.25093712131391843891172943852, −6.90245929966587211773087324747, −6.63266880715944236957990870622, −6.30911588357971738443301632702, −6.24580594389555839620556704293, −5.44364695187002043838865580903, −5.07351394520457083353807688653, −4.79017613349203602808028306780, −4.64065634741312102357396661084, −3.65164115354405144094532866704, −3.46704534565565412969661382740, −2.95144313672453620983700957428, −2.92134042800006823419670745599, −1.67584530289778924637150065375, −1.43034676041880394595126755240, 0, 0, 0,
1.43034676041880394595126755240, 1.67584530289778924637150065375, 2.92134042800006823419670745599, 2.95144313672453620983700957428, 3.46704534565565412969661382740, 3.65164115354405144094532866704, 4.64065634741312102357396661084, 4.79017613349203602808028306780, 5.07351394520457083353807688653, 5.44364695187002043838865580903, 6.24580594389555839620556704293, 6.30911588357971738443301632702, 6.63266880715944236957990870622, 6.90245929966587211773087324747, 7.25093712131391843891172943852, 7.41768819056655327024411483281, 7.86195856353724085969307969484, 7.955365973166934187778379192542, 8.424268292688031193921047813833, 8.546423511461009828464401955907, 8.758256292235516698379198558779, 9.571202944388799864132972588980, 9.709876464886236275882495479204, 9.882691801179596796962295591839, 10.24632024213934150814357778698
