| L(s) = 1 | + 4·3-s − 4·5-s − 7-s + 10·9-s − 11-s − 2·13-s − 16·15-s + 4·17-s + 7·19-s − 4·21-s + 23-s + 10·25-s + 20·27-s + 2·29-s − 4·31-s − 4·33-s + 4·35-s − 6·37-s − 8·39-s + 16·41-s + 5·43-s − 40·45-s − 17·49-s + 16·51-s + 11·53-s + 4·55-s + 28·57-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s − 0.377·7-s + 10/3·9-s − 0.301·11-s − 0.554·13-s − 4.13·15-s + 0.970·17-s + 1.60·19-s − 0.872·21-s + 0.208·23-s + 2·25-s + 3.84·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s + 0.676·35-s − 0.986·37-s − 1.28·39-s + 2.49·41-s + 0.762·43-s − 5.96·45-s − 2.42·49-s + 2.24·51-s + 1.51·53-s + 0.539·55-s + 3.70·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.80012927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.80012927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + T + 18 T^{2} + 13 T^{3} + 172 T^{4} + 13 p T^{5} + 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + T + 10 T^{2} + 13 T^{3} + 194 T^{4} + 13 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 22 T^{2} + 24 T^{3} + 342 T^{4} + 24 p T^{5} + 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 36 T^{2} - 224 T^{3} + 694 T^{4} - 224 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 60 T^{2} - 303 T^{3} + 1686 T^{4} - 303 p T^{5} + 60 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - T + 34 T^{2} - 37 T^{3} + 1070 T^{4} - 37 p T^{5} + 34 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 92 T^{2} - 172 T^{3} + 3678 T^{4} - 172 p T^{5} + 92 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 6 T + 130 T^{2} + 608 T^{3} + 6978 T^{4} + 608 p T^{5} + 130 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 5 T + 146 T^{2} - 481 T^{3} + 8690 T^{4} - 481 p T^{5} + 146 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 112 T^{2} - 92 T^{3} + 6726 T^{4} - 92 p T^{5} + 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 11 T + 244 T^{2} - 1773 T^{3} + 20250 T^{4} - 1773 p T^{5} + 244 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 6 T + 164 T^{2} - 796 T^{3} + 13714 T^{4} - 796 p T^{5} + 164 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 4 T + 164 T^{2} - 748 T^{3} + 13318 T^{4} - 748 p T^{5} + 164 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 12 T + 262 T^{2} - 2390 T^{3} + 26134 T^{4} - 2390 p T^{5} + 262 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 17 T + 290 T^{2} + 3331 T^{3} + 30600 T^{4} + 3331 p T^{5} + 290 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 3 T + 228 T^{2} - 5 p T^{3} + 22316 T^{4} - 5 p^{2} T^{5} + 228 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 3 T + 148 T^{2} + p T^{3} + 11746 T^{4} + p^{2} T^{5} + 148 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 10 T + 160 T^{2} - 1390 T^{3} + 12830 T^{4} - 1390 p T^{5} + 160 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 17 T + 420 T^{2} - 4397 T^{3} + 58116 T^{4} - 4397 p T^{5} + 420 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2 T + 196 T^{2} - 186 T^{3} + 19462 T^{4} - 186 p T^{5} + 196 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.73654220455693233574349618622, −5.10900472645700780290380264916, −5.07388780680986382623513801921, −4.87194105801586544989401206346, −4.83843594357390356478510918927, −4.46366553922253684668854776384, −4.39715860594401806443117503185, −4.15787504389829007082193780161, −3.95446950742149812172389910850, −3.54525175457924277954153308597, −3.53405750608004708519535069635, −3.49883359066363544506737844575, −3.40580888738522765585868884692, −3.06688858915689367860336259736, −2.74733592128433597633870629311, −2.73148620575302581181106812991, −2.64283551797617855945049853761, −2.13165338453468838603365355271, −1.96681361718451913361454680802, −1.73588337532180600710656249897, −1.65941859508523324258984640266, −0.930993008487317417094240965926, −0.855706236471068405835580097882, −0.67825156376149831247504244290, −0.45280914384978678153324353018,
0.45280914384978678153324353018, 0.67825156376149831247504244290, 0.855706236471068405835580097882, 0.930993008487317417094240965926, 1.65941859508523324258984640266, 1.73588337532180600710656249897, 1.96681361718451913361454680802, 2.13165338453468838603365355271, 2.64283551797617855945049853761, 2.73148620575302581181106812991, 2.74733592128433597633870629311, 3.06688858915689367860336259736, 3.40580888738522765585868884692, 3.49883359066363544506737844575, 3.53405750608004708519535069635, 3.54525175457924277954153308597, 3.95446950742149812172389910850, 4.15787504389829007082193780161, 4.39715860594401806443117503185, 4.46366553922253684668854776384, 4.83843594357390356478510918927, 4.87194105801586544989401206346, 5.07388780680986382623513801921, 5.10900472645700780290380264916, 5.73654220455693233574349618622
