L(s) = 1 | + 2·3-s − 4-s − 6·7-s + 9-s − 2·12-s − 3·16-s − 4·19-s − 12·21-s − 2·25-s − 4·27-s + 6·28-s + 6·31-s − 36-s − 4·37-s − 6·43-s − 6·48-s + 14·49-s − 8·57-s − 8·61-s − 6·63-s + 7·64-s + 4·67-s − 2·73-s − 4·75-s + 4·76-s − 4·79-s − 11·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1/2·4-s − 2.26·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.917·19-s − 2.61·21-s − 2/5·25-s − 0.769·27-s + 1.13·28-s + 1.07·31-s − 1/6·36-s − 0.657·37-s − 0.914·43-s − 0.866·48-s + 2·49-s − 1.05·57-s − 1.02·61-s − 0.755·63-s + 7/8·64-s + 0.488·67-s − 0.234·73-s − 0.461·75-s + 0.458·76-s − 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 73 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.663477846467172549479061889828, −9.396152038274767786766575447619, −8.945834811056945865824819305285, −8.422569047301544181563986321416, −8.026898874215042644635425441906, −7.16362665857073601207380088132, −6.70054833293196549592528243866, −6.27685355807958818406186977112, −5.64843217071871487921675422735, −4.67343670052114048280588496244, −4.03655616886073847473994412666, −3.36502920699158770798746585282, −2.94080839143346445109060006209, −2.11550072184673889865559323192, 0,
2.11550072184673889865559323192, 2.94080839143346445109060006209, 3.36502920699158770798746585282, 4.03655616886073847473994412666, 4.67343670052114048280588496244, 5.64843217071871487921675422735, 6.27685355807958818406186977112, 6.70054833293196549592528243866, 7.16362665857073601207380088132, 8.026898874215042644635425441906, 8.422569047301544181563986321416, 8.945834811056945865824819305285, 9.396152038274767786766575447619, 9.663477846467172549479061889828