| L(s) = 1 | + 2-s + 2·3-s − 10·4-s − 14·5-s + 2·6-s − 36·7-s + 7·8-s − 49·9-s − 14·10-s − 10·11-s − 20·12-s − 22·13-s − 36·14-s − 28·15-s + 61·16-s − 49·18-s + 22·19-s + 140·20-s − 72·21-s − 10·22-s − 380·23-s + 14·24-s + 37·25-s − 22·26-s + 46·27-s + 360·28-s + 78·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.384·3-s − 5/4·4-s − 1.25·5-s + 0.136·6-s − 1.94·7-s + 0.309·8-s − 1.81·9-s − 0.442·10-s − 0.274·11-s − 0.481·12-s − 0.469·13-s − 0.687·14-s − 0.481·15-s + 0.953·16-s − 0.641·18-s + 0.265·19-s + 1.56·20-s − 0.748·21-s − 0.0969·22-s − 3.44·23-s + 0.119·24-s + 0.295·25-s − 0.165·26-s + 0.327·27-s + 2.42·28-s + 0.499·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + 11 T^{2} - 7 p^{2} T^{3} + 21 p^{2} T^{4} - 7 p^{5} T^{5} + 11 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 53 T^{2} - 250 T^{3} + 163 p^{2} T^{4} - 250 p^{3} T^{5} + 53 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 14 T + 159 T^{2} - 1576 T^{3} - 20291 T^{4} - 1576 p^{3} T^{5} + 159 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 36 T + 831 T^{2} + 5778 T^{3} + 58279 T^{4} + 5778 p^{3} T^{5} + 831 p^{6} T^{6} + 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10 T + 3568 T^{2} + 40200 T^{3} + 6271361 T^{4} + 40200 p^{3} T^{5} + 3568 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 22 T + 5279 T^{2} + 164240 T^{3} + 13515729 T^{4} + 164240 p^{3} T^{5} + 5279 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 22 T + 18245 T^{2} - 624006 T^{3} + 160564655 T^{4} - 624006 p^{3} T^{5} + 18245 p^{6} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 380 T + 85875 T^{2} + 13106270 T^{3} + 1616857555 T^{4} + 13106270 p^{3} T^{5} + 85875 p^{6} T^{6} + 380 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 78 T + 94851 T^{2} - 5618376 T^{3} + 3436887625 T^{4} - 5618376 p^{3} T^{5} + 94851 p^{6} T^{6} - 78 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 362 T + 152360 T^{2} + 32276128 T^{3} + 7264494729 T^{4} + 32276128 p^{3} T^{5} + 152360 p^{6} T^{6} + 362 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 512 T + 190856 T^{2} + 50680960 T^{3} + 12444415950 T^{4} + 50680960 p^{3} T^{5} + 190856 p^{6} T^{6} + 512 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 840 T + 522636 T^{2} + 4959288 p T^{3} + 63721821766 T^{4} + 4959288 p^{4} T^{5} + 522636 p^{6} T^{6} + 840 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 114 T + 161220 T^{2} + 11775000 T^{3} + 17345138341 T^{4} + 11775000 p^{3} T^{5} + 161220 p^{6} T^{6} + 114 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 10 T + 329476 T^{2} + 4191888 T^{3} + 46702622069 T^{4} + 4191888 p^{3} T^{5} + 329476 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 50 T + 223267 T^{2} + 23693064 T^{3} + 25976703257 T^{4} + 23693064 p^{3} T^{5} + 223267 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 996 T + 958284 T^{2} - 541470132 T^{3} + 301264321318 T^{4} - 541470132 p^{3} T^{5} + 958284 p^{6} T^{6} - 996 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 448 T + 528746 T^{2} + 191029344 T^{3} + 163720487563 T^{4} + 191029344 p^{3} T^{5} + 528746 p^{6} T^{6} + 448 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 868 T + 962944 T^{2} + 605911940 T^{3} + 428616419390 T^{4} + 605911940 p^{3} T^{5} + 962944 p^{6} T^{6} + 868 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 1116 T + 1798284 T^{2} + 1235308700 T^{3} + 1031640282550 T^{4} + 1235308700 p^{3} T^{5} + 1798284 p^{6} T^{6} + 1116 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 540 T + 1193894 T^{2} + 347390120 T^{3} + 579821039015 T^{4} + 347390120 p^{3} T^{5} + 1193894 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 940 T + 932284 T^{2} + 641029836 T^{3} + 662196409398 T^{4} + 641029836 p^{3} T^{5} + 932284 p^{6} T^{6} + 940 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 850 T + 1884977 T^{2} - 1471992022 T^{3} + 1503367753131 T^{4} - 1471992022 p^{3} T^{5} + 1884977 p^{6} T^{6} - 850 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 784 T + 2377308 T^{2} - 1342609264 T^{3} + 2382598755046 T^{4} - 1342609264 p^{3} T^{5} + 2377308 p^{6} T^{6} - 784 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 518 T + 274919 T^{2} + 5296380 T^{3} + 1036759053865 T^{4} + 5296380 p^{3} T^{5} + 274919 p^{6} T^{6} - 518 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.631497477324700614442224030894, −8.309643766961813161497171890242, −8.151123025019016515318299641886, −8.144556316714523104268523669334, −7.52949493756290906091143460718, −7.36624280141174808768248155716, −7.32970152747774403772024660141, −6.68841852307096518509149672052, −6.49848290981917626391094187866, −6.48557347056441048287673871602, −5.72469144501147768294171428534, −5.70575266300603326491195040899, −5.59554661164460924808695712358, −5.04932019408226277636307018085, −4.80759154398092360658069768454, −4.60706308865243867456412133820, −4.08812214672394281205152864829, −3.78045879569925759419261907650, −3.66627462049416597051783225193, −3.29320207280255932527622863554, −3.26112264527392909397665587489, −2.71431644193917300017297487567, −2.22701234519216994959160723419, −1.85954986848823420319817747141, −1.26153218601736440864435994908, 0, 0, 0, 0,
1.26153218601736440864435994908, 1.85954986848823420319817747141, 2.22701234519216994959160723419, 2.71431644193917300017297487567, 3.26112264527392909397665587489, 3.29320207280255932527622863554, 3.66627462049416597051783225193, 3.78045879569925759419261907650, 4.08812214672394281205152864829, 4.60706308865243867456412133820, 4.80759154398092360658069768454, 5.04932019408226277636307018085, 5.59554661164460924808695712358, 5.70575266300603326491195040899, 5.72469144501147768294171428534, 6.48557347056441048287673871602, 6.49848290981917626391094187866, 6.68841852307096518509149672052, 7.32970152747774403772024660141, 7.36624280141174808768248155716, 7.52949493756290906091143460718, 8.144556316714523104268523669334, 8.151123025019016515318299641886, 8.309643766961813161497171890242, 8.631497477324700614442224030894
