| L(s) = 1 | − 16·11-s + 8·13-s − 16-s + 20·17-s − 20·23-s − 8·25-s + 32·31-s − 48·41-s − 12·43-s − 20·47-s − 8·67-s + 14·81-s + 24·83-s + 36·97-s + 8·101-s − 36·103-s − 48·107-s + 116·121-s + 127-s + 131-s + 137-s + 139-s − 128·143-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4.82·11-s + 2.21·13-s − 1/4·16-s + 4.85·17-s − 4.17·23-s − 8/5·25-s + 5.74·31-s − 7.49·41-s − 1.82·43-s − 2.91·47-s − 0.977·67-s + 14/9·81-s + 2.63·83-s + 3.65·97-s + 0.796·101-s − 3.54·103-s − 4.64·107-s + 10.5·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5087849001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5087849001\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 1294 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.094629354658803420988745614473, −7.966949031932369193881891996047, −7.84589382009722754857249333004, −7.79414139429600593637780892335, −7.22267585827641568428322435734, −6.61535021826330573440236830483, −6.39588803614483845672849229797, −6.29776237429667155659789947250, −6.27822416842115402773220930161, −5.72050041536769906331092709983, −5.33432562851094761789494607335, −5.31088522474708337891079453525, −5.14980432561787772450831420707, −4.95197989284377843706270277186, −4.65475771352574560352280897667, −3.90631357692479930178536478427, −3.75415106349849940817914898747, −3.32038582781388535323135462404, −3.28756673497654838920593747001, −2.87449900030571154165815576195, −2.76094361607858545763032868598, −1.89510258415444756031424330602, −1.79954589487102156967742689247, −1.30918334681467185394194591335, −0.25656437065591258210781784916,
0.25656437065591258210781784916, 1.30918334681467185394194591335, 1.79954589487102156967742689247, 1.89510258415444756031424330602, 2.76094361607858545763032868598, 2.87449900030571154165815576195, 3.28756673497654838920593747001, 3.32038582781388535323135462404, 3.75415106349849940817914898747, 3.90631357692479930178536478427, 4.65475771352574560352280897667, 4.95197989284377843706270277186, 5.14980432561787772450831420707, 5.31088522474708337891079453525, 5.33432562851094761789494607335, 5.72050041536769906331092709983, 6.27822416842115402773220930161, 6.29776237429667155659789947250, 6.39588803614483845672849229797, 6.61535021826330573440236830483, 7.22267585827641568428322435734, 7.79414139429600593637780892335, 7.84589382009722754857249333004, 7.966949031932369193881891996047, 8.094629354658803420988745614473
