L(s) = 1 | − 2·2-s + 3·4-s − 2·8-s − 4·11-s + 2·13-s − 8·17-s + 8·22-s − 8·23-s + 2·25-s − 4·26-s + 4·29-s + 8·31-s + 6·32-s + 16·34-s − 8·37-s + 16·41-s − 8·43-s − 12·44-s + 16·46-s − 12·47-s + 6·49-s − 4·50-s + 6·52-s + 8·53-s − 8·58-s + 4·59-s − 4·61-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.707·8-s − 1.20·11-s + 0.554·13-s − 1.94·17-s + 1.70·22-s − 1.66·23-s + 2/5·25-s − 0.784·26-s + 0.742·29-s + 1.43·31-s + 1.06·32-s + 2.74·34-s − 1.31·37-s + 2.49·41-s − 1.21·43-s − 1.80·44-s + 2.35·46-s − 1.75·47-s + 6/7·49-s − 0.565·50-s + 0.832·52-s + 1.09·53-s − 1.05·58-s + 0.520·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4229174757\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4229174757\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - 6 T^{2} - 13 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 16 T + 118 T^{2} - 896 T^{3} + 6867 T^{4} - 896 p T^{5} + 118 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} - 128 T^{3} + 299 T^{4} - 128 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 627 T^{4} + 48 p T^{5} + 46 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 74 T^{2} + 112 T^{3} + 3675 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 496 T^{3} - 4693 T^{4} - 496 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} + 64 T^{3} + 11387 T^{4} + 64 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T - 122 T^{2} - 112 T^{3} + 10827 T^{4} - 112 p T^{5} - 122 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 150 T^{2} + 112 T^{3} + 16595 T^{4} + 112 p T^{5} - 150 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.87756404174636934388452725605, −6.82972353973987186182441090836, −6.76286949829648400343269299873, −6.60129117118731816614613495282, −6.43844227735935604311757530123, −6.08107236841990545254918199923, −5.75422438953719674943869610389, −5.40398201171085981756693674085, −5.39219762465197733562509482306, −5.31410083578635015084590251709, −4.62734526353551417925264808755, −4.43949448631132940744958792583, −4.26748003410696829813051199102, −4.19353237577099871765797875559, −3.96884718914453844739361039552, −3.31424573775062588356958554739, −2.87704425225240092414644083667, −2.87619352380695748619738671982, −2.80867729586125448215695420271, −2.09382885740172134262185321631, −1.96418247978228395550318465927, −1.83632405039345651938622835395, −1.22031943869551928386609970268, −0.842263656801649286196721327915, −0.21849837017514945989382904470,
0.21849837017514945989382904470, 0.842263656801649286196721327915, 1.22031943869551928386609970268, 1.83632405039345651938622835395, 1.96418247978228395550318465927, 2.09382885740172134262185321631, 2.80867729586125448215695420271, 2.87619352380695748619738671982, 2.87704425225240092414644083667, 3.31424573775062588356958554739, 3.96884718914453844739361039552, 4.19353237577099871765797875559, 4.26748003410696829813051199102, 4.43949448631132940744958792583, 4.62734526353551417925264808755, 5.31410083578635015084590251709, 5.39219762465197733562509482306, 5.40398201171085981756693674085, 5.75422438953719674943869610389, 6.08107236841990545254918199923, 6.43844227735935604311757530123, 6.60129117118731816614613495282, 6.76286949829648400343269299873, 6.82972353973987186182441090836, 6.87756404174636934388452725605