| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 2·9-s + 8·11-s + 2·13-s − 4·15-s − 4·17-s − 2·19-s + 4·21-s + 4·23-s − 2·25-s + 6·27-s + 12·29-s + 16·33-s − 4·35-s + 4·37-s + 4·39-s − 4·41-s − 4·45-s − 8·47-s + 3·49-s − 8·51-s − 16·55-s − 4·57-s + 14·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 2.41·11-s + 0.554·13-s − 1.03·15-s − 0.970·17-s − 0.458·19-s + 0.872·21-s + 0.834·23-s − 2/5·25-s + 1.15·27-s + 2.22·29-s + 2.78·33-s − 0.676·35-s + 0.657·37-s + 0.640·39-s − 0.624·41-s − 0.596·45-s − 1.16·47-s + 3/7·49-s − 1.12·51-s − 2.15·55-s − 0.529·57-s + 1.82·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.787665294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.787665294\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.315674211653326294645218594033, −8.806835172112355571789658007072, −8.632774866063494430817119255835, −8.513010759144983044552780965438, −8.053216340371400571841393031486, −7.64581856985470557026708555669, −7.08029706683350936777743534686, −6.63039384518596305894554324957, −6.52150010985878345998611546336, −6.16099088283009944142005901262, −5.22451497337483502071372921225, −4.66964054502891255940766725085, −4.55543857223913567351642932973, −3.94086117614780184701889055717, −3.58393422774124097100238498128, −3.34408856227961594942603801579, −2.42653077716523436188038705623, −2.14882292726780365030061664787, −1.23789394600805257032243393900, −0.906383485706870872564815430846,
0.906383485706870872564815430846, 1.23789394600805257032243393900, 2.14882292726780365030061664787, 2.42653077716523436188038705623, 3.34408856227961594942603801579, 3.58393422774124097100238498128, 3.94086117614780184701889055717, 4.55543857223913567351642932973, 4.66964054502891255940766725085, 5.22451497337483502071372921225, 6.16099088283009944142005901262, 6.52150010985878345998611546336, 6.63039384518596305894554324957, 7.08029706683350936777743534686, 7.64581856985470557026708555669, 8.053216340371400571841393031486, 8.513010759144983044552780965438, 8.632774866063494430817119255835, 8.806835172112355571789658007072, 9.315674211653326294645218594033
