| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−10.23233687889639886881720044925, −9.730985256268271424878295911920, −9.019660204875550419148497591382, −8.364955351789872564250717727993, −8.215028652014338750632346686889, −7.19250214882024512254378722525, −6.74848060717112410656014338963, −6.09553790955098515565552989321, −5.91562815270661005160626581722, −4.97153580512610098325521001803, −4.52827833300760369208621410126, −3.84252961453441711169051333538, −2.76608702409746176095302424855, −1.75139797086746412243454574915, 0,
1.75139797086746412243454574915, 2.76608702409746176095302424855, 3.84252961453441711169051333538, 4.52827833300760369208621410126, 4.97153580512610098325521001803, 5.91562815270661005160626581722, 6.09553790955098515565552989321, 6.74848060717112410656014338963, 7.19250214882024512254378722525, 8.215028652014338750632346686889, 8.364955351789872564250717727993, 9.019660204875550419148497591382, 9.730985256268271424878295911920, 10.23233687889639886881720044925
