| L(s) = 1 | + 2·3-s + 9-s + 12·19-s − 2·25-s − 4·27-s + 4·43-s − 10·49-s + 24·57-s + 4·67-s − 12·73-s − 4·75-s − 11·81-s − 20·97-s + 14·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 20·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 12·171-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 2.75·19-s − 2/5·25-s − 0.769·27-s + 0.609·43-s − 1.42·49-s + 3.17·57-s + 0.488·67-s − 1.40·73-s − 0.461·75-s − 1.22·81-s − 2.03·97-s + 1.27·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.917·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.897359453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897359453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07809525398202980406044336160, −9.639016940320800334052280885364, −9.436600088083830005749484082288, −8.767180667207632935752662796440, −8.272753207478764876685966833970, −7.64609849831734536930680103253, −7.44100999727324721185445643986, −6.72427936360411292249661208927, −5.84525714091965020127006922055, −5.39740284302558434141147680580, −4.64687933809762396534139192468, −3.73995554294415330699335680159, −3.20983270019662921836084913625, −2.58879204967609884000331595226, −1.43228927579727342661843394247,
1.43228927579727342661843394247, 2.58879204967609884000331595226, 3.20983270019662921836084913625, 3.73995554294415330699335680159, 4.64687933809762396534139192468, 5.39740284302558434141147680580, 5.84525714091965020127006922055, 6.72427936360411292249661208927, 7.44100999727324721185445643986, 7.64609849831734536930680103253, 8.272753207478764876685966833970, 8.767180667207632935752662796440, 9.436600088083830005749484082288, 9.639016940320800334052280885364, 10.07809525398202980406044336160
