| L(s) = 1 | + 6·3-s − 21·7-s − 10·9-s − 6·11-s − 8·13-s + 52·17-s − 152·19-s − 126·21-s + 48·23-s − 4·27-s − 468·29-s − 268·31-s − 36·33-s − 262·37-s − 48·39-s − 74·41-s + 692·43-s + 774·47-s + 294·49-s + 312·51-s − 62·53-s − 912·57-s − 1.08e3·59-s + 174·61-s + 210·63-s − 212·67-s + 288·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.13·7-s − 0.370·9-s − 0.164·11-s − 0.170·13-s + 0.741·17-s − 1.83·19-s − 1.30·21-s + 0.435·23-s − 0.0285·27-s − 2.99·29-s − 1.55·31-s − 0.189·33-s − 1.16·37-s − 0.197·39-s − 0.281·41-s + 2.45·43-s + 2.40·47-s + 6/7·49-s + 0.856·51-s − 0.160·53-s − 2.11·57-s − 2.40·59-s + 0.365·61-s + 0.419·63-s − 0.386·67-s + 0.502·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 p T + 46 T^{2} - 332 T^{3} + 46 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 1138 T^{2} - 8648 T^{3} + 1138 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 3056 T^{2} + 4274 p T^{3} + 3056 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 52 T + 556 p T^{2} - 521086 T^{3} + 556 p^{4} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 p T + 6157 T^{2} - 2640 T^{3} + 6157 p^{3} T^{4} + 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 48 T + 15649 T^{2} + 301792 T^{3} + 15649 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 468 T + 112852 T^{2} + 19590970 T^{3} + 112852 p^{3} T^{4} + 468 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 268 T + 3251 p T^{2} + 15045992 T^{3} + 3251 p^{4} T^{4} + 268 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 262 T + 99139 T^{2} + 15491556 T^{3} + 99139 p^{3} T^{4} + 262 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 74 T + 148867 T^{2} + 6978388 T^{3} + 148867 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 692 T + 333157 T^{2} - 113597928 T^{3} + 333157 p^{3} T^{4} - 692 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 774 T + 470506 T^{2} - 169228476 T^{3} + 470506 p^{3} T^{4} - 774 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 62 T + 363751 T^{2} + 24260908 T^{3} + 363751 p^{3} T^{4} + 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1088 T + 930505 T^{2} + 466405760 T^{3} + 930505 p^{3} T^{4} + 1088 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 174 T + 132487 T^{2} + 108451156 T^{3} + 132487 p^{3} T^{4} - 174 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 212 T + 599425 T^{2} + 169097592 T^{3} + 599425 p^{3} T^{4} + 212 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1056 T + 831813 T^{2} + 484957632 T^{3} + 831813 p^{3} T^{4} + 1056 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 194 T + 720375 T^{2} - 231245436 T^{3} + 720375 p^{3} T^{4} - 194 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 362 T + 559502 T^{2} + 5683900 T^{3} + 559502 p^{3} T^{4} - 362 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1696 T + 2447313 T^{2} - 2029546432 T^{3} + 2447313 p^{3} T^{4} - 1696 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 158 T + 258707 T^{2} + 637693636 T^{3} + 258707 p^{3} T^{4} - 158 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1620 T + 1192836 T^{2} + 576628578 T^{3} + 1192836 p^{3} T^{4} + 1620 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.844997784138708198619489761624, −8.274772028190214633550473582746, −7.952746876124527869429729480082, −7.64736832828657830475412922409, −7.46509068884147060605557994843, −7.45413971854402702567585960453, −6.84052180489191872826242902439, −6.64318520114974435467349772855, −6.42942152315999937740430817200, −5.95225176175114510888573846261, −5.64204702589760174109966582950, −5.61824893152828389962084273007, −5.31567875629333997444065344568, −4.82192389268761233047941399740, −4.35453772826963661176063731069, −4.13350809488716331931541559385, −3.66820450778590921382144094644, −3.56657134502783987532903798370, −3.37892863125092973359305293831, −2.72405940432991029147432147777, −2.55032117664237834234459406135, −2.35968654519072704587375627897, −1.97761894515279846548055074928, −1.31851974997948560392666045974, −1.09793675630724318564419017165, 0, 0, 0,
1.09793675630724318564419017165, 1.31851974997948560392666045974, 1.97761894515279846548055074928, 2.35968654519072704587375627897, 2.55032117664237834234459406135, 2.72405940432991029147432147777, 3.37892863125092973359305293831, 3.56657134502783987532903798370, 3.66820450778590921382144094644, 4.13350809488716331931541559385, 4.35453772826963661176063731069, 4.82192389268761233047941399740, 5.31567875629333997444065344568, 5.61824893152828389962084273007, 5.64204702589760174109966582950, 5.95225176175114510888573846261, 6.42942152315999937740430817200, 6.64318520114974435467349772855, 6.84052180489191872826242902439, 7.45413971854402702567585960453, 7.46509068884147060605557994843, 7.64736832828657830475412922409, 7.952746876124527869429729480082, 8.274772028190214633550473582746, 8.844997784138708198619489761624
