| L(s) = 1 | − 4·5-s − 2·11-s + 4·13-s − 6·17-s − 4·19-s + 4·23-s + 4·25-s − 3·29-s − 14·31-s − 2·37-s − 10·41-s − 6·43-s + 2·47-s − 21·49-s + 8·55-s − 8·59-s + 2·61-s − 16·65-s + 20·67-s − 12·71-s + 2·73-s − 30·79-s − 32·83-s + 24·85-s − 10·89-s + 16·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.603·11-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 4/5·25-s − 0.557·29-s − 2.51·31-s − 0.328·37-s − 1.56·41-s − 0.914·43-s + 0.291·47-s − 3·49-s + 1.07·55-s − 1.04·59-s + 0.256·61-s − 1.98·65-s + 2.44·67-s − 1.42·71-s + 0.234·73-s − 3.37·79-s − 3.51·83-s + 2.60·85-s − 1.05·89-s + 1.64·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 6 p T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 4 T^{2} - 36 T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 20 T^{2} - 102 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 120 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 152 T^{2} + 936 T^{3} + 152 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 79 T^{2} + 116 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 59 T^{2} + 308 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 92 T^{2} + 548 T^{3} + 92 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 24 T^{2} + 264 T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 116 T^{2} - 58 T^{3} + 116 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 173 T^{2} + 928 T^{3} + 173 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 83 T^{2} + 84 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 20 T + 233 T^{2} - 2040 T^{3} + 233 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 65 T^{2} + 8 T^{3} + 65 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 123 T^{2} - 452 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 30 T + 488 T^{2} + 5128 T^{3} + 488 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 32 T + 565 T^{2} + 6288 T^{3} + 565 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 11 T^{2} - 1036 T^{3} + 11 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 323 T^{2} + 2652 T^{3} + 323 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.581196080501614564061624647491, −8.154303598590062384287749711524, −7.986931709816135725205925670570, −7.890279297595819422487669867731, −7.34229928945729248462755651243, −7.06899140319896616235224883613, −7.02794935031423803730159247547, −6.80017534694693748230683961031, −6.41474317325176001010741641115, −6.19703319688135327179743465892, −5.72479697020013955130265098143, −5.42330705680848296647383354654, −5.35557823044645516243132333029, −4.88653855231884743113592302929, −4.46998002674105200197250844760, −4.46186212320372177533266659220, −3.84122964550264485555890387985, −3.78216224203900691998719050149, −3.71310453412188766425006018070, −3.08013811145037779912101720989, −2.85085532493849969305992105017, −2.57619967267814222924903421468, −1.81761337770903729443900394781, −1.56846702150384281821622001011, −1.40793790101020764370838205946, 0, 0, 0,
1.40793790101020764370838205946, 1.56846702150384281821622001011, 1.81761337770903729443900394781, 2.57619967267814222924903421468, 2.85085532493849969305992105017, 3.08013811145037779912101720989, 3.71310453412188766425006018070, 3.78216224203900691998719050149, 3.84122964550264485555890387985, 4.46186212320372177533266659220, 4.46998002674105200197250844760, 4.88653855231884743113592302929, 5.35557823044645516243132333029, 5.42330705680848296647383354654, 5.72479697020013955130265098143, 6.19703319688135327179743465892, 6.41474317325176001010741641115, 6.80017534694693748230683961031, 7.02794935031423803730159247547, 7.06899140319896616235224883613, 7.34229928945729248462755651243, 7.890279297595819422487669867731, 7.986931709816135725205925670570, 8.154303598590062384287749711524, 8.581196080501614564061624647491
