| L(s) = 1 | − 3-s − 2·9-s + 7·11-s + 5·13-s − 17-s − 4·19-s + 6·23-s − 3·27-s + 3·29-s − 10·31-s − 7·33-s − 18·37-s − 5·39-s − 18·41-s − 9·47-s + 51-s − 10·53-s + 4·57-s − 6·59-s − 24·61-s + 10·67-s − 6·69-s + 4·71-s − 6·73-s + 17·79-s + 8·81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 2.11·11-s + 1.38·13-s − 0.242·17-s − 0.917·19-s + 1.25·23-s − 0.577·27-s + 0.557·29-s − 1.79·31-s − 1.21·33-s − 2.95·37-s − 0.800·39-s − 2.81·41-s − 1.31·47-s + 0.140·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s − 3.07·61-s + 1.22·67-s − 0.722·69-s + 0.474·71-s − 0.702·73-s + 1.91·79-s + 8/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 27 T^{2} + 54 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 43 T^{2} + 160 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 540 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 18 T + 191 T^{2} + 1388 T^{3} + 191 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 89 T^{2} - 64 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 93 T^{2} + 614 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 115 T^{2} + 588 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 365 T^{2} + 3368 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 137 T^{2} - 828 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 740 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 205 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 107 T^{2} + 1168 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 95 T^{2} + 464 T^{3} + 95 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 275 T^{2} + 1702 T^{3} + 275 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.84761595638645795457229111836, −6.83060634260935490854640187992, −6.63086223981356413318771171877, −6.52124817589914402903675413361, −6.14602813777282541664424761077, −6.12693725892238562898070439152, −5.83193424866580079453443764379, −5.47309886478975727750237411707, −5.21680982471079136986284722023, −5.12498263918387015289198724300, −4.88061085036576032263737559258, −4.61207834480145202676600263138, −4.28145919347132189166717194685, −3.99542494216663733990644151862, −3.67229130997970809654876111203, −3.66823415505613817931506098735, −3.21976239501058502223547587228, −3.14329748324406158330848040812, −3.10251448019834493031888156929, −2.19463629093603385689219371487, −2.15073016546840975710269930548, −1.73670139120075347509436197335, −1.50584760255655438128973000450, −1.26997559335188338874629758040, −1.09446416799895602715214706682, 0, 0, 0,
1.09446416799895602715214706682, 1.26997559335188338874629758040, 1.50584760255655438128973000450, 1.73670139120075347509436197335, 2.15073016546840975710269930548, 2.19463629093603385689219371487, 3.10251448019834493031888156929, 3.14329748324406158330848040812, 3.21976239501058502223547587228, 3.66823415505613817931506098735, 3.67229130997970809654876111203, 3.99542494216663733990644151862, 4.28145919347132189166717194685, 4.61207834480145202676600263138, 4.88061085036576032263737559258, 5.12498263918387015289198724300, 5.21680982471079136986284722023, 5.47309886478975727750237411707, 5.83193424866580079453443764379, 6.12693725892238562898070439152, 6.14602813777282541664424761077, 6.52124817589914402903675413361, 6.63086223981356413318771171877, 6.83060634260935490854640187992, 6.84761595638645795457229111836
