| L(s) = 1 | + 3·9-s + 4·11-s − 4·19-s − 6·29-s − 18·31-s + 2·41-s + 28·49-s − 8·59-s + 16·61-s + 46·71-s + 4·79-s − 7·81-s + 4·89-s + 12·99-s + 16·101-s − 36·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 31·169-s − 12·171-s + ⋯ |
| L(s) = 1 | + 9-s + 1.20·11-s − 0.917·19-s − 1.11·29-s − 3.23·31-s + 0.312·41-s + 4·49-s − 1.04·59-s + 2.04·61-s + 5.45·71-s + 0.450·79-s − 7/9·81-s + 0.423·89-s + 1.20·99-s + 1.59·101-s − 3.44·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.38·169-s − 0.917·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.863281035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.863281035\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 766 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 51 T^{2} + 440 T^{4} - 51 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 232 T^{2} + 22366 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 23 T + 270 T^{2} - 23 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 71 T^{2} + 864 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7830 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 19806 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.68313702753329900727565561084, −5.60552835906618092184507567650, −5.51122781450979889972416021916, −5.46396278588778067804249992353, −5.21115867951101085044547258072, −4.85209705010492440072095838169, −4.55329600351763514626853545604, −4.51560405082236760485936610681, −4.21486758469910550504300723464, −3.98123345621833431160936344067, −3.84303442951403467757731697900, −3.77619244735410390382509217800, −3.64983264246631906651278477697, −3.35194577839899351848530003943, −2.95049883687882608160636197925, −2.90537976812905827439564027603, −2.31593614068389934968889262050, −2.24743638525632042165945964563, −1.94821887379017217620657539035, −1.83800231383341328446544915782, −1.81316568199707686160866374596, −1.16740541872437630584034084022, −0.814681270084052902003881604503, −0.800811225401481398194365439291, −0.30738558575493389002729804043,
0.30738558575493389002729804043, 0.800811225401481398194365439291, 0.814681270084052902003881604503, 1.16740541872437630584034084022, 1.81316568199707686160866374596, 1.83800231383341328446544915782, 1.94821887379017217620657539035, 2.24743638525632042165945964563, 2.31593614068389934968889262050, 2.90537976812905827439564027603, 2.95049883687882608160636197925, 3.35194577839899351848530003943, 3.64983264246631906651278477697, 3.77619244735410390382509217800, 3.84303442951403467757731697900, 3.98123345621833431160936344067, 4.21486758469910550504300723464, 4.51560405082236760485936610681, 4.55329600351763514626853545604, 4.85209705010492440072095838169, 5.21115867951101085044547258072, 5.46396278588778067804249992353, 5.51122781450979889972416021916, 5.60552835906618092184507567650, 5.68313702753329900727565561084
