| L(s) = 1 | − 2-s − 2·3-s − 5-s + 2·6-s − 7-s + 8-s + 3·9-s + 10-s + 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s + 23-s − 2·24-s + 25-s − 2·26-s − 4·27-s − 2·30-s + 35-s − 2·38-s − 4·39-s − 40-s − 2·42-s − 3·45-s + ⋯ |
| L(s) = 1 | − 2-s − 2·3-s − 5-s + 2·6-s − 7-s + 8-s + 3·9-s + 10-s + 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s + 23-s − 2·24-s + 25-s − 2·26-s − 4·27-s − 2·30-s + 35-s − 2·38-s − 4·39-s − 40-s − 2·42-s − 3·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1920983180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1920983180\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.53807098707566870139151734339, −10.84714600543848557037994163102, −10.42221649036897612790423638392, −10.30356323187396066346087723628, −9.596514782753109281686101974481, −9.182627850665603609226991013567, −8.914048376752615677525062219855, −8.145240894306979817949908777756, −7.61066734956533661391529047058, −7.37309771885186120907951493767, −6.67027968540024341763391217447, −6.51818548568021431076483180974, −5.86682746942423310554254338530, −5.17581826413679925562563802003, −5.01052437586629966409980113166, −4.03627033499675448547696490701, −3.82233050635133160964287254791, −3.09464739201220127694230549691, −1.38591687451132591132221097641, −0.881950195852900869598048203446,
0.881950195852900869598048203446, 1.38591687451132591132221097641, 3.09464739201220127694230549691, 3.82233050635133160964287254791, 4.03627033499675448547696490701, 5.01052437586629966409980113166, 5.17581826413679925562563802003, 5.86682746942423310554254338530, 6.51818548568021431076483180974, 6.67027968540024341763391217447, 7.37309771885186120907951493767, 7.61066734956533661391529047058, 8.145240894306979817949908777756, 8.914048376752615677525062219855, 9.182627850665603609226991013567, 9.596514782753109281686101974481, 10.30356323187396066346087723628, 10.42221649036897612790423638392, 10.84714600543848557037994163102, 11.53807098707566870139151734339
