| L(s) = 1 | + 3·2-s + 9·3-s − 8·4-s − 15·5-s + 27·6-s + 34·7-s − 28·8-s + 30·9-s − 45·10-s + 52·11-s − 72·12-s + 19·13-s + 102·14-s − 135·15-s + 57·16-s − 51·17-s + 90·18-s − 153·19-s + 120·20-s + 306·21-s + 156·22-s + 162·23-s − 252·24-s + 150·25-s + 57·26-s + 43·27-s − 272·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.73·3-s − 4-s − 1.34·5-s + 1.83·6-s + 1.83·7-s − 1.23·8-s + 10/9·9-s − 1.42·10-s + 1.42·11-s − 1.73·12-s + 0.405·13-s + 1.94·14-s − 2.32·15-s + 0.890·16-s − 0.727·17-s + 1.17·18-s − 1.84·19-s + 1.34·20-s + 3.17·21-s + 1.51·22-s + 1.46·23-s − 2.14·24-s + 6/5·25-s + 0.429·26-s + 0.306·27-s − 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.413428033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.413428033\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 47 T^{3} + 17 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - p^{2} T + 17 p T^{2} - 232 T^{3} + 17 p^{4} T^{4} - p^{8} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 34 T + 1079 T^{2} - 23280 T^{3} + 1079 p^{3} T^{4} - 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 52 T + 4255 T^{2} - 131640 T^{3} + 4255 p^{3} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 19 T + 3503 T^{2} - 63078 T^{3} + 3503 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 153 T + 24705 T^{2} + 2055742 T^{3} + 24705 p^{3} T^{4} + 153 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 162 T + 33959 T^{2} - 3354848 T^{3} + 33959 p^{3} T^{4} - 162 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 45 T + 41855 T^{2} - 3758830 T^{3} + 41855 p^{3} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 67 T + 90455 T^{2} + 3992628 T^{3} + 90455 p^{3} T^{4} + 67 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 308 T + 167747 T^{2} + 30322920 T^{3} + 167747 p^{3} T^{4} + 308 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 498 T + 254751 T^{2} - 67696460 T^{3} + 254751 p^{3} T^{4} - 498 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 246 T + 181205 T^{2} + 31053676 T^{3} + 181205 p^{3} T^{4} + 246 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 101 T + 57549 T^{2} - 58547970 T^{3} + 57549 p^{3} T^{4} - 101 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 893 T + 701847 T^{2} - 288997110 T^{3} + 701847 p^{3} T^{4} - 893 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 355 T + 550657 T^{2} - 122030058 T^{3} + 550657 p^{3} T^{4} - 355 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1019 T + 1002319 T^{2} - 494848778 T^{3} + 1002319 p^{3} T^{4} - 1019 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 334 T + 908949 T^{2} - 198667084 T^{3} + 908949 p^{3} T^{4} - 334 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 313 T + 220279 T^{2} - 328713084 T^{3} + 220279 p^{3} T^{4} - 313 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 639 T + 561171 T^{2} - 520727922 T^{3} + 561171 p^{3} T^{4} - 639 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 92 T + 473747 T^{2} + 401620008 T^{3} + 473747 p^{3} T^{4} + 92 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2736 T + 4202357 T^{2} - 3879938416 T^{3} + 4202357 p^{3} T^{4} - 2736 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1623 T + 2707547 T^{2} - 2262712862 T^{3} + 2707547 p^{3} T^{4} - 1623 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 475 T + 1815539 T^{2} + 393329682 T^{3} + 1815539 p^{3} T^{4} + 475 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−12.57742948317407991377594083788, −11.96700065379150229602644614104, −11.76637313414445777803365983016, −11.26615596369562402083065683204, −10.87340721492894986045430752165, −10.72665792889003534743245662327, −10.10753785302168163968277595760, −9.179278558063589002750104557368, −9.131126785841765477617157557679, −8.860029766187203479054809222653, −8.385150797751017728678597587354, −8.300035359640679909893669673492, −7.909279614510233861968010838681, −7.37500117205015938657276090202, −6.75505135012596944977442189398, −6.51524584481201277119234762981, −5.37220924724512324166708663936, −5.02466628915192005655822693514, −4.70009792045002992450189535737, −4.00385662350735805975507730683, −3.87795527397625336367432038680, −3.70063376827350616544234187635, −2.62531486020739504467690275918, −1.88011937806202983031938187515, −0.844794642058816184049789383884,
0.844794642058816184049789383884, 1.88011937806202983031938187515, 2.62531486020739504467690275918, 3.70063376827350616544234187635, 3.87795527397625336367432038680, 4.00385662350735805975507730683, 4.70009792045002992450189535737, 5.02466628915192005655822693514, 5.37220924724512324166708663936, 6.51524584481201277119234762981, 6.75505135012596944977442189398, 7.37500117205015938657276090202, 7.909279614510233861968010838681, 8.300035359640679909893669673492, 8.385150797751017728678597587354, 8.860029766187203479054809222653, 9.131126785841765477617157557679, 9.179278558063589002750104557368, 10.10753785302168163968277595760, 10.72665792889003534743245662327, 10.87340721492894986045430752165, 11.26615596369562402083065683204, 11.76637313414445777803365983016, 11.96700065379150229602644614104, 12.57742948317407991377594083788
