| L(s) = 1 | − 3·5-s + 3·11-s + 4·13-s − 6·17-s − 2·19-s − 2·23-s + 6·25-s − 6·29-s + 4·31-s + 6·37-s − 6·41-s + 2·43-s + 14·47-s − 7·49-s − 6·53-s − 9·55-s + 4·59-s − 12·65-s + 12·67-s + 10·71-s − 6·73-s + 4·79-s + 4·83-s + 18·85-s − 12·89-s + 6·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.417·23-s + 6/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.304·43-s + 2.04·47-s − 49-s − 0.824·53-s − 1.21·55-s + 0.520·59-s − 1.48·65-s + 1.46·67-s + 1.18·71-s − 0.702·73-s + 0.450·79-s + 0.439·83-s + 1.95·85-s − 1.27·89-s + 0.615·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.045012035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.045012035\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + p T^{2} - 16 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + p T^{2} - 8 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 49 T^{2} + 168 T^{3} + 49 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 60 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 45 T^{2} + 76 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 216 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 83 T^{2} - 16 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 1300 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 115 T^{2} + 404 T^{3} + 115 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 157 T^{2} - 440 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 45 T^{2} - 16 T^{3} - 45 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 193 T^{2} - 1576 T^{3} + 193 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 121 T^{2} - 1492 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 200 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 117 T^{2} - 920 T^{3} + 117 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T - 21 T^{2} + 1168 T^{3} - 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 1260 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.08588538246036496353085511327, −6.57565914071588866255000517095, −6.46952288873415835745020589509, −6.42923900041193145096099218526, −6.00607031873828036620076040268, −5.83294363793922668496261532781, −5.65182422483677650554724055030, −5.09463510357571939565624763040, −4.94769682880040317224161619617, −4.92186902152704414789994707432, −4.28713143687582927553303061746, −4.21440329507926158668823124887, −4.09962227094541030688644284897, −3.82555965982813357286118015104, −3.67989048443947794201973746108, −3.31197152145623431429711060555, −3.02496804240663165493719755301, −2.62448910976994393767144653487, −2.53773993258718925036281094637, −1.99230321558861440120664656278, −1.69169927610135160158495235336, −1.59549443629513187607759390112, −0.927976871389569294018248088191, −0.59546545722128343648614875853, −0.40568118235162382630963376134,
0.40568118235162382630963376134, 0.59546545722128343648614875853, 0.927976871389569294018248088191, 1.59549443629513187607759390112, 1.69169927610135160158495235336, 1.99230321558861440120664656278, 2.53773993258718925036281094637, 2.62448910976994393767144653487, 3.02496804240663165493719755301, 3.31197152145623431429711060555, 3.67989048443947794201973746108, 3.82555965982813357286118015104, 4.09962227094541030688644284897, 4.21440329507926158668823124887, 4.28713143687582927553303061746, 4.92186902152704414789994707432, 4.94769682880040317224161619617, 5.09463510357571939565624763040, 5.65182422483677650554724055030, 5.83294363793922668496261532781, 6.00607031873828036620076040268, 6.42923900041193145096099218526, 6.46952288873415835745020589509, 6.57565914071588866255000517095, 7.08588538246036496353085511327
