| L(s) = 1 | + 2·2-s + 4-s + 4·7-s − 2·8-s − 4·13-s + 8·14-s − 4·16-s − 4·19-s − 12·23-s − 20·25-s − 8·26-s + 4·28-s − 10·31-s − 2·32-s + 8·37-s − 8·38-s + 6·41-s − 4·43-s − 24·46-s + 18·47-s − 2·49-s − 40·50-s − 4·52-s − 8·56-s − 4·61-s − 20·62-s + 3·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.707·8-s − 1.10·13-s + 2.13·14-s − 16-s − 0.917·19-s − 2.50·23-s − 4·25-s − 1.56·26-s + 0.755·28-s − 1.79·31-s − 0.353·32-s + 1.31·37-s − 1.29·38-s + 0.937·41-s − 0.609·43-s − 3.53·46-s + 2.62·47-s − 2/7·49-s − 5.65·50-s − 0.554·52-s − 1.06·56-s − 0.512·61-s − 2.54·62-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7972440790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7972440790\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 56 T^{3} - 233 T^{4} - 56 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 56 T^{3} - 89 T^{4} - 56 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 40 T^{2} + 759 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T + 17 T^{2} + 378 T^{3} - 2796 T^{4} + 378 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 272 T^{3} - 2213 T^{4} - 272 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 167 T^{2} - 1134 T^{3} + 7212 T^{4} - 1134 p T^{5} + 167 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 92 T^{2} - 56 T^{3} + 6967 T^{4} - 56 p T^{5} - 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} + 16 T^{3} + 8647 T^{4} + 16 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24 T + 284 T^{2} - 3024 T^{3} + 30567 T^{4} - 3024 p T^{5} + 284 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 79 T^{2} + 378 T^{3} + 2100 T^{4} + 378 p T^{5} - 79 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 74 T^{2} + 448 T^{3} + 3427 T^{4} + 448 p T^{5} - 74 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.07050301650544773169553773833, −6.60674306128165652978239806337, −6.52406943404557237396144454896, −6.09638356597467066913215482113, −5.98922418081981259733355221532, −5.94486942680931660687090481416, −5.66182230502224107988673588182, −5.41637994831004954911996578516, −5.17306533412580048168666889105, −4.93179569527938255578046857822, −4.83635718095075239160703070015, −4.37829126499696644239218707552, −4.11800526393798678339728959162, −4.06367321453607835245713154791, −3.99226403441310973087061179695, −3.61401955531830710289242884803, −3.48959090219732976851824148163, −2.97887351528297223984012302770, −2.46965044179967238599455400720, −2.18343041074378861882633693891, −2.12626020948935320056648264552, −2.01722857757644649047580231820, −1.58069310904531437747395411289, −0.875607131160451955133827264559, −0.15300582918808224398793566783,
0.15300582918808224398793566783, 0.875607131160451955133827264559, 1.58069310904531437747395411289, 2.01722857757644649047580231820, 2.12626020948935320056648264552, 2.18343041074378861882633693891, 2.46965044179967238599455400720, 2.97887351528297223984012302770, 3.48959090219732976851824148163, 3.61401955531830710289242884803, 3.99226403441310973087061179695, 4.06367321453607835245713154791, 4.11800526393798678339728959162, 4.37829126499696644239218707552, 4.83635718095075239160703070015, 4.93179569527938255578046857822, 5.17306533412580048168666889105, 5.41637994831004954911996578516, 5.66182230502224107988673588182, 5.94486942680931660687090481416, 5.98922418081981259733355221532, 6.09638356597467066913215482113, 6.52406943404557237396144454896, 6.60674306128165652978239806337, 7.07050301650544773169553773833
