| L(s) = 1 | + 3-s − 3·5-s + 7-s − 4·9-s − 11·13-s − 3·15-s − 3·17-s − 3·19-s + 21-s + 9·23-s + 6·25-s − 5·27-s − 7·29-s − 6·31-s − 3·35-s − 20·37-s − 11·39-s − 22·41-s + 10·43-s + 12·45-s − 4·49-s − 3·51-s − 7·53-s − 3·57-s − 11·59-s − 16·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s − 4/3·9-s − 3.05·13-s − 0.774·15-s − 0.727·17-s − 0.688·19-s + 0.218·21-s + 1.87·23-s + 6/5·25-s − 0.962·27-s − 1.29·29-s − 1.07·31-s − 0.507·35-s − 3.28·37-s − 1.76·39-s − 3.43·41-s + 1.52·43-s + 1.78·45-s − 4/7·49-s − 0.420·51-s − 0.961·53-s − 0.397·57-s − 1.43·59-s − 2.04·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - 4 T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - 30 T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 55 T^{2} + 200 T^{3} + 55 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 98 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 89 T^{2} - 422 T^{3} + 89 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 43 T^{2} + 114 T^{3} + 43 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 237 T^{2} + 1724 T^{3} + 237 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $D_{6}$ | \( 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 97 T^{2} - 508 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} + 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T - 9 T^{2} - 600 T^{3} - 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 11 T + 37 T^{2} - 246 T^{3} + 37 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 101 T^{2} - 396 T^{3} + 101 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 47 T^{2} - 498 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 445 T^{2} + 4604 T^{3} + 445 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 297 T^{2} - 2340 T^{3} + 297 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 188 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 233 T^{2} - 1260 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.860802365675483176211066258883, −8.540145357664334987504622043863, −8.254614497709732878824615339164, −8.150540179535444430516104060189, −7.59823200099215991281733525146, −7.38394875407072967812688464763, −7.35935278243352838393312261647, −7.08762236184424338004813353648, −6.76096379848190048134819747909, −6.57726309160933334728241837472, −6.03341621095946184504670611890, −5.63045405788185112398923498702, −5.41099517224066101967811834690, −4.99732562136610174913424466559, −4.79066872583948236119717666634, −4.78464883701679355126920616205, −4.40055200700155236134090076653, −3.69124099594208146294265936297, −3.61041505369148414840700250199, −3.25534928703236262790507389285, −2.80590639723048745176077481032, −2.78532423606824347924427876538, −2.21360812421070456407622505263, −1.72229109001735573255998072950, −1.54513954007592035166355729424, 0, 0, 0,
1.54513954007592035166355729424, 1.72229109001735573255998072950, 2.21360812421070456407622505263, 2.78532423606824347924427876538, 2.80590639723048745176077481032, 3.25534928703236262790507389285, 3.61041505369148414840700250199, 3.69124099594208146294265936297, 4.40055200700155236134090076653, 4.78464883701679355126920616205, 4.79066872583948236119717666634, 4.99732562136610174913424466559, 5.41099517224066101967811834690, 5.63045405788185112398923498702, 6.03341621095946184504670611890, 6.57726309160933334728241837472, 6.76096379848190048134819747909, 7.08762236184424338004813353648, 7.35935278243352838393312261647, 7.38394875407072967812688464763, 7.59823200099215991281733525146, 8.150540179535444430516104060189, 8.254614497709732878824615339164, 8.540145357664334987504622043863, 8.860802365675483176211066258883
