| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9225076033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9225076033\) |
| \(L(\frac{3}{2})\) |
|
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
−6.17269631997449473083014008692, −5.67926337164662786597921635712, −5.54643467118831597964147160140, −5.48529711124727722453989382501, −5.33855694215167573424426150401, −4.85589532939490264404353267197, −4.78847061464987600296706832343, −4.71411692283797849182570394181, −4.55895239649700015890287367839, −4.10802198924701935031267741934, −4.02953152613130550176963609214, −4.00025442006870157291092781782, −3.90864825721463456899853527256, −3.17552935359166450437443391236, −3.14811503521687225275132233019, −3.01105649629125168353490350917, −2.89145620261105331716836119729, −2.17090770541049017705128978192, −1.83125422038797786114867377836, −1.81506559997777465915650910921, −1.77484091816106355446527456177, −0.808566214391868846920907958231, −0.78795161604112686557859108059, −0.71666207466301013753571570161, −0.27954295091045972860612423197,
0.27954295091045972860612423197, 0.71666207466301013753571570161, 0.78795161604112686557859108059, 0.808566214391868846920907958231, 1.77484091816106355446527456177, 1.81506559997777465915650910921, 1.83125422038797786114867377836, 2.17090770541049017705128978192, 2.89145620261105331716836119729, 3.01105649629125168353490350917, 3.14811503521687225275132233019, 3.17552935359166450437443391236, 3.90864825721463456899853527256, 4.00025442006870157291092781782, 4.02953152613130550176963609214, 4.10802198924701935031267741934, 4.55895239649700015890287367839, 4.71411692283797849182570394181, 4.78847061464987600296706832343, 4.85589532939490264404353267197, 5.33855694215167573424426150401, 5.48529711124727722453989382501, 5.54643467118831597964147160140, 5.67926337164662786597921635712, 6.17269631997449473083014008692
