| L(s) = 1 | + 3-s + 3·5-s + 2·7-s + 9-s + 7·11-s − 13-s + 3·15-s + 10·17-s − 13·19-s + 2·21-s + 3·23-s + 6·25-s − 27-s + 13·29-s − 8·31-s + 7·33-s + 6·35-s + 5·37-s − 39-s + 8·41-s + 24·43-s + 3·45-s + 2·47-s − 9·49-s + 10·51-s + 53-s + 21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.755·7-s + 1/3·9-s + 2.11·11-s − 0.277·13-s + 0.774·15-s + 2.42·17-s − 2.98·19-s + 0.436·21-s + 0.625·23-s + 6/5·25-s − 0.192·27-s + 2.41·29-s − 1.43·31-s + 1.21·33-s + 1.01·35-s + 0.821·37-s − 0.160·39-s + 1.24·41-s + 3.65·43-s + 0.447·45-s + 0.291·47-s − 9/7·49-s + 1.40·51-s + 0.137·53-s + 2.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.086237373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.086237373\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 27 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 40 T^{2} - 140 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 16 T^{2} - 24 T^{3} + 16 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 61 T^{2} - 257 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 13 T + 90 T^{2} + 432 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 113 T^{2} - 678 T^{3} + 113 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 105 T^{2} + 495 T^{3} + 105 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 5 T + 19 T^{2} + 126 T^{3} + 19 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 49 T^{2} + 212 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 59 T^{2} - 406 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 17 T + 243 T^{2} + 2042 T^{3} + 243 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 132 T^{2} - 866 T^{3} + 132 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 109 T^{2} - 174 T^{3} + 109 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 22 T + 309 T^{2} + 2899 T^{3} + 309 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 368 T^{3} + 43 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 120 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 289 T^{2} + 2454 T^{3} + 289 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 210 T^{2} + 632 T^{3} + 210 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.029566655532443377009804811765, −8.778106974065532144620875794308, −8.620498439568672473218523814249, −8.210824853956913487815980141276, −7.70804397715753483771551032933, −7.54378579353636160191221604105, −7.46680651700850437000737249647, −6.93154349237669741552996497672, −6.45344847258585934584209080982, −6.41275931831062217971523831169, −6.01611649839586153297714950037, −5.90556893088585129142762414397, −5.58319440601672552522792286112, −5.01946537443459750564044220193, −4.69221658477154116300660982004, −4.35912673720096826832977216483, −4.24402798863863742446540885582, −3.61485315379859829599376351663, −3.56742418908269208095011621392, −2.66584819461882898238006159893, −2.64312412631501903512489538871, −2.21276668962332384358640616607, −1.58709543811419562535628536652, −1.26999445059091199403677573027, −0.947027120518869112509197893180,
0.947027120518869112509197893180, 1.26999445059091199403677573027, 1.58709543811419562535628536652, 2.21276668962332384358640616607, 2.64312412631501903512489538871, 2.66584819461882898238006159893, 3.56742418908269208095011621392, 3.61485315379859829599376351663, 4.24402798863863742446540885582, 4.35912673720096826832977216483, 4.69221658477154116300660982004, 5.01946537443459750564044220193, 5.58319440601672552522792286112, 5.90556893088585129142762414397, 6.01611649839586153297714950037, 6.41275931831062217971523831169, 6.45344847258585934584209080982, 6.93154349237669741552996497672, 7.46680651700850437000737249647, 7.54378579353636160191221604105, 7.70804397715753483771551032933, 8.210824853956913487815980141276, 8.620498439568672473218523814249, 8.778106974065532144620875794308, 9.029566655532443377009804811765
