| L(s) = 1 | + 9·2-s − 81·3-s − 159·4-s − 444·5-s − 729·6-s + 1.61e3·7-s − 1.00e3·8-s + 4.37e3·9-s − 3.99e3·10-s − 3.99e3·11-s + 1.28e4·12-s + 2.07e4·13-s + 1.45e4·14-s + 3.59e4·15-s + 2.16e4·16-s − 1.45e4·17-s + 3.93e4·18-s + 2.44e4·19-s + 7.05e4·20-s − 1.30e5·21-s − 3.59e4·22-s + 3.50e4·23-s + 8.15e4·24-s − 4.05e3·25-s + 1.86e5·26-s − 1.96e5·27-s − 2.56e5·28-s + ⋯ |
| L(s) = 1 | + 0.795·2-s − 1.73·3-s − 1.24·4-s − 1.58·5-s − 1.37·6-s + 1.77·7-s − 0.695·8-s + 2·9-s − 1.26·10-s − 0.904·11-s + 2.15·12-s + 2.62·13-s + 1.41·14-s + 2.75·15-s + 1.32·16-s − 0.717·17-s + 1.59·18-s + 0.819·19-s + 1.97·20-s − 3.08·21-s − 0.719·22-s + 0.601·23-s + 1.20·24-s − 0.0519·25-s + 2.08·26-s − 1.92·27-s − 2.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.777279015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777279015\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + p^{3} T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 9 T + 15 p^{4} T^{2} - 323 p^{3} T^{3} + 15 p^{11} T^{4} - 9 p^{14} T^{5} + p^{21} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 444 T + 40239 p T^{2} + 2735768 p^{2} T^{3} + 40239 p^{8} T^{4} + 444 p^{14} T^{5} + p^{21} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 1614 T + 307233 T^{2} + 790656308 T^{3} + 307233 p^{7} T^{4} - 1614 p^{14} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 20772 T + 232682259 T^{2} - 1941065209864 T^{3} + 232682259 p^{7} T^{4} - 20772 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 14538 T + 412756467 T^{2} + 200146212980 T^{3} + 412756467 p^{7} T^{4} + 14538 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 24492 T + 1476867765 T^{2} - 49394356429992 T^{3} + 1476867765 p^{7} T^{4} - 24492 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 35094 T + 2262349653 T^{2} - 38515282207028 T^{3} + 2262349653 p^{7} T^{4} - 35094 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 179862 T + 40311941247 T^{2} + 6266602437157788 T^{3} + 40311941247 p^{7} T^{4} + 179862 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 288888 T + 77083618461 T^{2} - 13983720499847440 T^{3} + 77083618461 p^{7} T^{4} - 288888 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 107562 T + 273599913795 T^{2} - 19232280372286428 T^{3} + 273599913795 p^{7} T^{4} - 107562 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 135198 T + 575331993483 T^{2} + 51384550416508124 T^{3} + 575331993483 p^{7} T^{4} + 135198 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 193536 T + 633525664605 T^{2} - 125885577366905616 T^{3} + 633525664605 p^{7} T^{4} - 193536 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 591486 T + 908422267245 T^{2} + 693984302571198916 T^{3} + 908422267245 p^{7} T^{4} + 591486 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 79044 T + 2529330228603 T^{2} + 108778384552517688 T^{3} + 2529330228603 p^{7} T^{4} - 79044 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2532768 T + 6588293215113 T^{2} - 8761483458577361984 T^{3} + 6588293215113 p^{7} T^{4} - 2532768 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6678792 T + 21974604832851 T^{2} - 46509029370020353824 T^{3} + 21974604832851 p^{7} T^{4} - 6678792 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 7150356 T + 35053255038225 T^{2} - 99829660769282829176 T^{3} + 35053255038225 p^{7} T^{4} - 7150356 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1390398 T + 6393443942565 T^{2} - 11581466543393663652 T^{3} + 6393443942565 p^{7} T^{4} - 1390398 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6429114 T + 44280311135751 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + 44280311135751 p^{7} T^{4} + 6429114 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6873186 T + 51506564223993 T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + 51506564223993 p^{7} T^{4} - 6873186 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6505596 T + 71173099306545 T^{2} - \)\(27\!\cdots\!56\)\( T^{3} + 71173099306545 p^{7} T^{4} - 6505596 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8842962 T + 131349994025895 T^{2} + \)\(75\!\cdots\!48\)\( T^{3} + 131349994025895 p^{7} T^{4} + 8842962 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1764774 T + 235749361482351 T^{2} + \)\(28\!\cdots\!08\)\( T^{3} + 235749361482351 p^{7} T^{4} + 1764774 p^{14} T^{5} + p^{21} T^{6} \) |
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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
−13.53965830041610961169063305447, −12.99594857733780843944513069647, −12.96186728818168817648073684278, −12.30487448005142951221979649038, −11.60983804246546302531416540358, −11.38504644380779954265468307027, −11.22067432079704780392344798186, −11.03814588654630380630178365596, −10.26430602955328430821602256398, −9.905434998401344222267219897997, −8.976245712437801136349338021334, −8.418633978411514535834731843957, −8.219846220628577470480357523172, −7.73795537709544526707996100357, −7.18492424403679019216604719904, −6.46621421868459614613733885279, −5.62005528668941755079597499147, −5.24529004688261846788062933561, −5.06677085602354374016002269664, −4.15874559777591029348866659882, −4.07821256261418294211626003997, −3.65699150972471508719795582775, −1.83221538782156362695084312582, −0.74013886104304199327858929722, −0.70835910546680713600633167677,
0.70835910546680713600633167677, 0.74013886104304199327858929722, 1.83221538782156362695084312582, 3.65699150972471508719795582775, 4.07821256261418294211626003997, 4.15874559777591029348866659882, 5.06677085602354374016002269664, 5.24529004688261846788062933561, 5.62005528668941755079597499147, 6.46621421868459614613733885279, 7.18492424403679019216604719904, 7.73795537709544526707996100357, 8.219846220628577470480357523172, 8.418633978411514535834731843957, 8.976245712437801136349338021334, 9.905434998401344222267219897997, 10.26430602955328430821602256398, 11.03814588654630380630178365596, 11.22067432079704780392344798186, 11.38504644380779954265468307027, 11.60983804246546302531416540358, 12.30487448005142951221979649038, 12.96186728818168817648073684278, 12.99594857733780843944513069647, 13.53965830041610961169063305447
