| L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·9-s + 4·11-s − 16-s + 4·18-s − 4·22-s − 4·23-s − 6·25-s − 6·29-s − 5·32-s + 4·36-s − 6·37-s + 4·43-s − 4·44-s + 4·46-s + 6·50-s + 18·53-s + 6·58-s + 7·64-s + 24·67-s + 8·71-s − 12·72-s + 6·74-s + 8·79-s + 7·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 4/3·9-s + 1.20·11-s − 1/4·16-s + 0.942·18-s − 0.852·22-s − 0.834·23-s − 6/5·25-s − 1.11·29-s − 0.883·32-s + 2/3·36-s − 0.986·37-s + 0.609·43-s − 0.603·44-s + 0.589·46-s + 0.848·50-s + 2.47·53-s + 0.787·58-s + 7/8·64-s + 2.93·67-s + 0.949·71-s − 1.41·72-s + 0.697·74-s + 0.900·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
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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
−8.255086570346816560791190073674, −7.927551709619254529663031492733, −7.38521269364537121491067114980, −6.87883629442846744538895773404, −6.41256441597080382535070806450, −5.82828256340678026582526594703, −5.39389191519356120871967358004, −5.11295536969314040608719834631, −4.08640154704599434295015231653, −3.91220472684603293138599057695, −3.49342612030087635821998425108, −2.41538718272791139781027122513, −1.97181873865615217959794017517, −0.997017127823322240793774924403, 0,
0.997017127823322240793774924403, 1.97181873865615217959794017517, 2.41538718272791139781027122513, 3.49342612030087635821998425108, 3.91220472684603293138599057695, 4.08640154704599434295015231653, 5.11295536969314040608719834631, 5.39389191519356120871967358004, 5.82828256340678026582526594703, 6.41256441597080382535070806450, 6.87883629442846744538895773404, 7.38521269364537121491067114980, 7.927551709619254529663031492733, 8.255086570346816560791190073674
