| L(s) = 1 | − 2-s − 2·8-s − 3·11-s + 2·13-s + 3·16-s − 2·17-s + 8·19-s + 3·22-s − 2·26-s + 10·29-s + 8·31-s − 3·32-s + 2·34-s + 6·37-s − 8·38-s + 14·41-s − 4·43-s − 8·47-s − 5·49-s − 6·53-s − 10·58-s − 12·59-s − 6·61-s − 8·62-s + 2·64-s + 4·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.707·8-s − 0.904·11-s + 0.554·13-s + 3/4·16-s − 0.485·17-s + 1.83·19-s + 0.639·22-s − 0.392·26-s + 1.85·29-s + 1.43·31-s − 0.530·32-s + 0.342·34-s + 0.986·37-s − 1.29·38-s + 2.18·41-s − 0.609·43-s − 1.16·47-s − 5/7·49-s − 0.824·53-s − 1.31·58-s − 1.56·59-s − 0.768·61-s − 1.01·62-s + 1/4·64-s + 0.488·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812046075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812046075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T - T^{2} - 116 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 167 T^{2} - 1156 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 56 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 644 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1096 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 153 T^{2} - 472 T^{3} + 153 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 181 T^{2} + 1008 T^{3} + 181 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 223 T^{2} - 1700 T^{3} + 223 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 173 T^{2} - 1096 T^{3} + 173 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} + 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 215 T^{2} - 1580 T^{3} + 215 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 399 T^{2} + 4276 T^{3} + 399 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.065425533008553259484504372142, −7.77329965653966197018633728137, −7.54487065004958476052937792256, −7.19234490876545850690681772870, −6.94768065661682225446732753205, −6.50347981109917767765377986394, −6.40665978546195629469866990454, −6.04825957625956048759198549546, −6.03986585962032891931000608521, −5.48876852697537827736456793678, −5.39933523579249686625526608453, −4.89951534323509872467481483669, −4.69813017101905642134716650735, −4.46113333627580420978602455270, −4.32036737446344089561028567884, −3.60873364680059656527923423887, −3.27444334529712552788000894132, −3.19152721554331398219805452977, −2.95132670489929551242914862052, −2.46696373462620922263472622611, −2.24929861665857050811999273130, −1.65192556245490859446944456158, −1.20110902874961205027213507930, −0.78022008218009118041254602973, −0.45525087172520054253600267706,
0.45525087172520054253600267706, 0.78022008218009118041254602973, 1.20110902874961205027213507930, 1.65192556245490859446944456158, 2.24929861665857050811999273130, 2.46696373462620922263472622611, 2.95132670489929551242914862052, 3.19152721554331398219805452977, 3.27444334529712552788000894132, 3.60873364680059656527923423887, 4.32036737446344089561028567884, 4.46113333627580420978602455270, 4.69813017101905642134716650735, 4.89951534323509872467481483669, 5.39933523579249686625526608453, 5.48876852697537827736456793678, 6.03986585962032891931000608521, 6.04825957625956048759198549546, 6.40665978546195629469866990454, 6.50347981109917767765377986394, 6.94768065661682225446732753205, 7.19234490876545850690681772870, 7.54487065004958476052937792256, 7.77329965653966197018633728137, 8.065425533008553259484504372142
