| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 2·5-s + 3·6-s − 10·8-s − 6·10-s + 11-s − 6·12-s − 13-s − 2·15-s + 15·16-s + 12·20-s − 3·22-s + 10·24-s + 25-s + 3·26-s + 6·30-s − 22·32-s − 33-s + 39-s − 20·40-s + 2·41-s − 2·43-s + 6·44-s + 2·47-s − 15·48-s + ⋯ |
| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 2·5-s + 3·6-s − 10·8-s − 6·10-s + 11-s − 6·12-s − 13-s − 2·15-s + 15·16-s + 12·20-s − 3·22-s + 10·24-s + 25-s + 3·26-s + 6·30-s − 22·32-s − 33-s + 39-s − 20·40-s + 2·41-s − 2·43-s + 6·44-s + 2·47-s − 15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1387495708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1387495708\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 5 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{8} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−8.460915175321761235443975782793, −8.267743014963939036285783666670, −7.80364615787510018936144533052, −7.73551708271388336413552364642, −7.20908087491782886748791381401, −7.20877837705509464941076740476, −6.88690813531367525138184837542, −6.65782931712861500635309174840, −6.41294804193249137275589038306, −6.25518012263535103592175532361, −6.07804249518703855234572348331, −5.54001912774411378084669632374, −5.43197034432738427027019327933, −5.34198176815464359448850852181, −5.27858051588762843407143305543, −4.42425776941067545086850175368, −3.97555379983719071504035548865, −3.59862104190831343971752432541, −3.40355059272266370336791755239, −2.73249382940162996800094119073, −2.27925826571073484171743575601, −2.18858965324182768959442761440, −2.16217725319526669289692058913, −1.27280001683882914027761634588, −1.11081059766122049202867734004,
1.11081059766122049202867734004, 1.27280001683882914027761634588, 2.16217725319526669289692058913, 2.18858965324182768959442761440, 2.27925826571073484171743575601, 2.73249382940162996800094119073, 3.40355059272266370336791755239, 3.59862104190831343971752432541, 3.97555379983719071504035548865, 4.42425776941067545086850175368, 5.27858051588762843407143305543, 5.34198176815464359448850852181, 5.43197034432738427027019327933, 5.54001912774411378084669632374, 6.07804249518703855234572348331, 6.25518012263535103592175532361, 6.41294804193249137275589038306, 6.65782931712861500635309174840, 6.88690813531367525138184837542, 7.20877837705509464941076740476, 7.20908087491782886748791381401, 7.73551708271388336413552364642, 7.80364615787510018936144533052, 8.267743014963939036285783666670, 8.460915175321761235443975782793
