| L(s) = 1 | − 660·13-s − 5.06e3·17-s − 5.54e3·37-s + 8.71e3·41-s + 9.56e4·53-s − 1.43e5·61-s + 8.64e4·73-s + 8.71e4·81-s + 2.46e3·97-s + 5.87e5·101-s − 4.68e4·113-s + 6.13e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.17e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.08·13-s − 4.24·17-s − 0.665·37-s + 0.809·41-s + 4.67·53-s − 4.93·61-s + 1.89·73-s + 1.47·81-s + 0.0265·97-s + 5.73·101-s − 0.344·113-s + 3.80·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.586·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.479602374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.479602374\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 87122 T^{4} + p^{20} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 549771682 T^{4} + p^{20} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 306602 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 330 T + 54450 T^{2} + 330 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2530 T + 3200450 T^{2} + 2530 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2549802 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 68424579426718 T^{4} + p^{20} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 34283082 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 222542 p T^{2} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2770 T + 3836450 T^{2} + 2770 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2178 T + p^{5} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 35368995141923122 T^{4} + p^{20} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 62763367693678078 T^{4} + p^{20} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 47830 T + 1143854450 T^{2} - 47830 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 693226598 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 35882 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 2056557036860651602 T^{4} + p^{20} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 894616798 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 43230 T + 934416450 T^{2} - 43230 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 5673984798 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 7276763020942840082 T^{4} + p^{20} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 1924732878 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1230 T + 756450 T^{2} - 1230 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23316512281745283489232443903, −6.98405665870692257804712361159, −6.91361833754589925361790239110, −6.54883129014935147662487027109, −6.49204734186441541860349634935, −5.89825983575944649431321144987, −5.86277668950416943455681574677, −5.73481602893949889458225910964, −5.15064703135633514759530460085, −4.80287331770110658221778125052, −4.70309250929625057908708920339, −4.41015798840778967627552723889, −4.35340241105729351608377147347, −4.04465533982118009816183851745, −3.48510024829449422319992825095, −3.19149509949423687517392965240, −3.10075915194474378097827084069, −2.32527892251211426172158101898, −2.17291549402563526941784270285, −2.17287152481114052290610149932, −1.98989794737957290897775853581, −1.31295783682318706234672943366, −0.70130318991100519062555050103, −0.41728810129350140301967538304, −0.39658871895481923981925215144,
0.39658871895481923981925215144, 0.41728810129350140301967538304, 0.70130318991100519062555050103, 1.31295783682318706234672943366, 1.98989794737957290897775853581, 2.17287152481114052290610149932, 2.17291549402563526941784270285, 2.32527892251211426172158101898, 3.10075915194474378097827084069, 3.19149509949423687517392965240, 3.48510024829449422319992825095, 4.04465533982118009816183851745, 4.35340241105729351608377147347, 4.41015798840778967627552723889, 4.70309250929625057908708920339, 4.80287331770110658221778125052, 5.15064703135633514759530460085, 5.73481602893949889458225910964, 5.86277668950416943455681574677, 5.89825983575944649431321144987, 6.49204734186441541860349634935, 6.54883129014935147662487027109, 6.91361833754589925361790239110, 6.98405665870692257804712361159, 7.23316512281745283489232443903
