| 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
| \(L(2)\) |
\(\approx\) |
\(8.522080152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.522080152\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2 \wr S_4$ | \( 1 + 518 T + 274919 T^{2} - 5296380 T^{3} + 1036759053865 T^{4} - 5296380 p^{3} T^{5} + 274919 p^{6} T^{6} + 518 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−8.235963758049758909855201282864, −7.73182356196945111656420594085, −7.69223027891475294621712511004, −7.53968919376169217809685064915, −7.25392217499869294897228868819, −6.55602655569628172647002928663, −6.43924335829186201567622937286, −6.34700321269914098712455093599, −5.83688354797005032744661284908, −5.49284447568122218835737753599, −5.30653330535734058924376189113, −5.27137445812745668473326672679, −4.90600437733807725950956989364, −4.72183095476753956642594272714, −4.32715981421542794272048262291, −4.19001783055156185261809165098, −3.73607373124081558355553741559, −3.23775744233268118687292169589, −2.73467890646142650795429502929, −2.57585420925917948519762422843, −2.20354503888185315116604074935, −1.77717183517194332236806717685, −1.04045155592535806375875552209, −0.792103463451985258247776784243, −0.68970907472670813519798433179,
0.68970907472670813519798433179, 0.792103463451985258247776784243, 1.04045155592535806375875552209, 1.77717183517194332236806717685, 2.20354503888185315116604074935, 2.57585420925917948519762422843, 2.73467890646142650795429502929, 3.23775744233268118687292169589, 3.73607373124081558355553741559, 4.19001783055156185261809165098, 4.32715981421542794272048262291, 4.72183095476753956642594272714, 4.90600437733807725950956989364, 5.27137445812745668473326672679, 5.30653330535734058924376189113, 5.49284447568122218835737753599, 5.83688354797005032744661284908, 6.34700321269914098712455093599, 6.43924335829186201567622937286, 6.55602655569628172647002928663, 7.25392217499869294897228868819, 7.53968919376169217809685064915, 7.69223027891475294621712511004, 7.73182356196945111656420594085, 8.235963758049758909855201282864
