L(s) = 1 | − 3·5-s + 3·7-s + 3·8-s − 6·11-s + 3·13-s − 3·17-s + 6·23-s − 3·29-s + 6·31-s − 9·35-s + 15·37-s − 9·40-s + 12·41-s + 12·43-s + 6·49-s − 12·53-s + 18·55-s + 9·56-s + 18·59-s − 3·61-s + 64-s − 9·65-s + 6·67-s + 9·73-s − 18·77-s + 6·79-s − 12·83-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 1.13·7-s + 1.06·8-s − 1.80·11-s + 0.832·13-s − 0.727·17-s + 1.25·23-s − 0.557·29-s + 1.07·31-s − 1.52·35-s + 2.46·37-s − 1.42·40-s + 1.87·41-s + 1.82·43-s + 6/7·49-s − 1.64·53-s + 2.42·55-s + 1.20·56-s + 2.34·59-s − 0.384·61-s + 1/8·64-s − 1.11·65-s + 0.733·67-s + 1.05·73-s − 2.05·77-s + 0.675·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.050028360\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.050028360\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
good | 2 | $D_{6}$ | \( 1 - 3 T^{3} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} + 18 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 126 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 27 T^{2} + 114 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 180 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 210 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 912 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 84 T^{2} - 380 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 117 T^{2} + 24 T^{3} + 117 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 12 T + 180 T^{2} + 1266 T^{3} + 180 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 189 T^{2} - 1572 T^{3} + 189 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 178 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 542 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 132 T^{2} + 108 T^{3} + 132 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1298 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 686 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1920 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 15 T + 315 T^{2} + 2622 T^{3} + 315 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 2144 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.645581392557475341830787099772, −9.117042056171947074786138601113, −8.991995863000387477360003426770, −8.675807900136106353411101632154, −8.049930774046966205195793375803, −7.997790217070732317057995708772, −7.88560486120336934818194137283, −7.55962259273529413884355354640, −7.51649042831801620093837149522, −6.89238860850333395018729279396, −6.62036596123719874308845812674, −6.17642581414690064612506196135, −5.75756433197239491036309687589, −5.45599049509064150458631604132, −5.03389189316331404711274933554, −4.82726169756478738766241634704, −4.24350905981333780661521834035, −4.17067308450487679386062782940, −4.03627440016763742496060989068, −3.25618973559653280192275619474, −2.70814742136531414936084382173, −2.51032438587698456512283239726, −1.97327481307346946432861525220, −1.13520287727974584344038223712, −0.69247686109260914487240933215,
0.69247686109260914487240933215, 1.13520287727974584344038223712, 1.97327481307346946432861525220, 2.51032438587698456512283239726, 2.70814742136531414936084382173, 3.25618973559653280192275619474, 4.03627440016763742496060989068, 4.17067308450487679386062782940, 4.24350905981333780661521834035, 4.82726169756478738766241634704, 5.03389189316331404711274933554, 5.45599049509064150458631604132, 5.75756433197239491036309687589, 6.17642581414690064612506196135, 6.62036596123719874308845812674, 6.89238860850333395018729279396, 7.51649042831801620093837149522, 7.55962259273529413884355354640, 7.88560486120336934818194137283, 7.997790217070732317057995708772, 8.049930774046966205195793375803, 8.675807900136106353411101632154, 8.991995863000387477360003426770, 9.117042056171947074786138601113, 9.645581392557475341830787099772