L(s) = 1 | − 5·2-s + 6·4-s − 4·5-s + 10·7-s + 10·8-s − 9-s + 20·10-s + 11-s − 20·13-s − 50·14-s − 10·16-s + 5·18-s + 25·19-s − 24·20-s − 5·22-s − 20·23-s + 25·25-s + 100·26-s + 20·27-s + 60·28-s − 40·29-s − 58·31-s − 85·32-s − 40·35-s − 6·36-s + 90·37-s − 125·38-s + ⋯ |
L(s) = 1 | − 5/2·2-s + 3/2·4-s − 4/5·5-s + 10/7·7-s + 5/4·8-s − 1/9·9-s + 2·10-s + 1/11·11-s − 1.53·13-s − 3.57·14-s − 5/8·16-s + 5/18·18-s + 1.31·19-s − 6/5·20-s − 0.227·22-s − 0.869·23-s + 25-s + 3.84·26-s + 0.740·27-s + 15/7·28-s − 1.37·29-s − 1.87·31-s − 2.65·32-s − 8/7·35-s − 1/6·36-s + 2.43·37-s − 3.28·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07219953010\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07219953010\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 11 | $C_4$ | \( 1 - T - 19 p T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
good | 2 | $C_2^2:C_4$ | \( 1 + 5 T + 19 T^{2} + 55 T^{3} + 121 T^{4} + 55 p^{2} T^{5} + 19 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + T^{2} - 20 T^{3} + 61 T^{4} - 20 p^{2} T^{5} + p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + 4 T - 9 T^{2} - 136 T^{3} - 319 T^{4} - 136 p^{2} T^{5} - 9 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 10 T + 129 T^{2} - 1310 T^{3} + 8361 T^{4} - 1310 p^{2} T^{5} + 129 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 20 T + 369 T^{2} + 6070 T^{3} + 81261 T^{4} + 6070 p^{2} T^{5} + 369 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + p^{2} T^{2} + 7800 T^{3} - 17879 T^{4} + 7800 p^{2} T^{5} + p^{6} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 25 T + 561 T^{2} - 5855 T^{3} + 141756 T^{4} - 5855 p^{2} T^{5} + 561 p^{4} T^{6} - 25 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 10 T + 1078 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 40 T + 1881 T^{2} + 81410 T^{3} + 2180301 T^{4} + 81410 p^{2} T^{5} + 1881 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 58 T + 423 T^{2} - 17434 T^{3} - 243175 T^{4} - 17434 p^{2} T^{5} + 423 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 90 T + 3491 T^{2} - 145800 T^{3} + 6716341 T^{4} - 145800 p^{2} T^{5} + 3491 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 80 T + 6401 T^{2} + 8520 p T^{3} + 9241 p^{2} T^{4} + 8520 p^{3} T^{5} + 6401 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 5771 T^{2} + 15084961 T^{4} - 5771 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 30 T - 1569 T^{2} - 98050 T^{3} + 581001 T^{4} - 98050 p^{2} T^{5} - 1569 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 120 T + 2591 T^{2} + 279810 T^{3} - 24164819 T^{4} + 279810 p^{2} T^{5} + 2591 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 23 T - 2427 T^{2} - 104641 T^{3} + 14691380 T^{4} - 104641 p^{2} T^{5} - 2427 p^{4} T^{6} - 23 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 10 T + 3601 T^{2} + 243430 T^{3} + 7253641 T^{4} + 243430 p^{2} T^{5} + 3601 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 115 T + 11923 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 148 T + 9423 T^{2} - 841046 T^{3} + 82451165 T^{4} - 841046 p^{2} T^{5} + 9423 p^{4} T^{6} - 148 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 300 T + 46429 T^{2} - 5018400 T^{3} + 415041241 T^{4} - 5018400 p^{2} T^{5} + 46429 p^{4} T^{6} - 300 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 70 T + 10021 T^{2} - 1216880 T^{3} + 75589621 T^{4} - 1216880 p^{2} T^{5} + 10021 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 225 T + 27589 T^{2} - 2589975 T^{3} + 221504596 T^{4} - 2589975 p^{2} T^{5} + 27589 p^{4} T^{6} - 225 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 61 T + 8161 T^{2} - 61 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 165 T + 7431 T^{2} + 999895 T^{3} + 180445176 T^{4} + 999895 p^{2} T^{5} + 7431 p^{4} T^{6} + 165 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.32613063810895925261879876950, −15.20094616670566885973205042721, −14.93121693169646358630650555256, −14.91052484225163681840645908379, −14.55693974802130800132633714662, −13.89344913458055276651457442668, −13.63415939960201137791630209978, −12.96619860766929219369018928180, −12.47605149836903241195723430943, −11.94778841066133950882930639167, −11.87472738208697920816204857607, −11.02728797715383661094363382787, −10.91802326774198080675652898897, −10.22897205180947550939390734720, −9.688702513640343836898559977297, −9.418783603637110004561332066081, −9.239798527400808300795729658819, −8.557336269125950525672707067218, −8.082194301962094467619559610771, −7.85159363374975635698652658666, −7.60075896120508420027768281865, −6.75128768761676733105855494842, −5.26043379360225753227016445339, −5.11354937382164602751352405843, −3.85750981494506242316922035002,
3.85750981494506242316922035002, 5.11354937382164602751352405843, 5.26043379360225753227016445339, 6.75128768761676733105855494842, 7.60075896120508420027768281865, 7.85159363374975635698652658666, 8.082194301962094467619559610771, 8.557336269125950525672707067218, 9.239798527400808300795729658819, 9.418783603637110004561332066081, 9.688702513640343836898559977297, 10.22897205180947550939390734720, 10.91802326774198080675652898897, 11.02728797715383661094363382787, 11.87472738208697920816204857607, 11.94778841066133950882930639167, 12.47605149836903241195723430943, 12.96619860766929219369018928180, 13.63415939960201137791630209978, 13.89344913458055276651457442668, 14.55693974802130800132633714662, 14.91052484225163681840645908379, 14.93121693169646358630650555256, 15.20094616670566885973205042721, 16.32613063810895925261879876950