| L(s) = 1 | − 4·3-s + 10·9-s − 8·11-s + 4·13-s − 4·17-s − 12·19-s − 4·23-s − 20·27-s + 8·29-s − 16·31-s + 32·33-s + 8·37-s − 16·39-s − 4·41-s − 4·43-s + 12·47-s − 12·49-s + 16·51-s + 48·57-s − 8·59-s + 12·61-s + 24·67-s + 16·69-s − 12·71-s + 24·73-s − 12·79-s + 35·81-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 2.41·11-s + 1.10·13-s − 0.970·17-s − 2.75·19-s − 0.834·23-s − 3.84·27-s + 1.48·29-s − 2.87·31-s + 5.57·33-s + 1.31·37-s − 2.56·39-s − 0.624·41-s − 0.609·43-s + 1.75·47-s − 1.71·49-s + 2.24·51-s + 6.35·57-s − 1.04·59-s + 1.53·61-s + 2.93·67-s + 1.92·69-s − 1.42·71-s + 2.80·73-s − 1.35·79-s + 35/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279018561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279018561\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 50 T^{2} + 240 T^{3} + 866 T^{4} + 240 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 24 T^{2} - 20 T^{3} + 14 T^{4} - 20 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 460 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 48 T^{2} + 52 T^{3} + 926 T^{4} + 52 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 100 T^{2} - 504 T^{3} + 4646 T^{4} - 504 p T^{5} + 100 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 150 T^{2} + 436 T^{3} + 8930 T^{4} + 436 p T^{5} + 150 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 104 T^{2} + 132 T^{3} + 4734 T^{4} + 132 p T^{5} + 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 158 T^{2} - 1180 T^{3} + 9410 T^{4} - 1180 p T^{5} + 158 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} + 160 T^{3} + 8870 T^{4} + 160 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 242 T^{2} + 1392 T^{3} + 21602 T^{4} + 1392 p T^{5} + 242 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 200 T^{2} - 1460 T^{3} + 15486 T^{4} - 1460 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 24 T + 404 T^{2} - 4888 T^{3} + 45030 T^{4} - 4888 p T^{5} + 404 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 238 T^{2} + 1980 T^{3} + 24226 T^{4} + 1980 p T^{5} + 238 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 24 T + 404 T^{2} - 4488 T^{3} + 43318 T^{4} - 4488 p T^{5} + 404 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 184 T^{2} + 1948 T^{3} + 22862 T^{4} + 1948 p T^{5} + 184 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 126 T^{2} - 540 T^{3} + 2754 T^{4} - 540 p T^{5} + 126 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 166 T^{2} - 220 T^{3} + 13890 T^{4} - 220 p T^{5} + 166 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 340 T^{2} - 1944 T^{3} + 47126 T^{4} - 1944 p T^{5} + 340 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.68651302433080708384897741846, −5.23131975916466351113149239450, −5.14133920779470504053840225042, −5.10669100397354025692682665061, −4.90216820413656520654561007067, −4.47810269452771585981505890072, −4.45423879598615738266760145735, −4.44461136302250517848205069694, −4.14403714492357139085008656529, −3.79695618634887057297061956725, −3.66681168140864451469881512575, −3.65624354485832973876511367936, −3.35298968027386971958932648160, −2.82235567713555755904745936845, −2.75316715297478678991346418440, −2.53427001903886427552440171945, −2.38475630607198249673502468579, −1.85148404412449192725283813141, −1.79002846068794539973371902259, −1.73893526077584047457498087662, −1.66647291057730089026257462286, −0.67310170329487669446547646675, −0.63666143068376451521559599950, −0.55274360259801295630814271448, −0.31707819442204257499110134278,
0.31707819442204257499110134278, 0.55274360259801295630814271448, 0.63666143068376451521559599950, 0.67310170329487669446547646675, 1.66647291057730089026257462286, 1.73893526077584047457498087662, 1.79002846068794539973371902259, 1.85148404412449192725283813141, 2.38475630607198249673502468579, 2.53427001903886427552440171945, 2.75316715297478678991346418440, 2.82235567713555755904745936845, 3.35298968027386971958932648160, 3.65624354485832973876511367936, 3.66681168140864451469881512575, 3.79695618634887057297061956725, 4.14403714492357139085008656529, 4.44461136302250517848205069694, 4.45423879598615738266760145735, 4.47810269452771585981505890072, 4.90216820413656520654561007067, 5.10669100397354025692682665061, 5.14133920779470504053840225042, 5.23131975916466351113149239450, 5.68651302433080708384897741846
