| L(s) = 1 | − 4·2-s + 10·4-s − 2·5-s − 20·8-s + 8·10-s + 2·11-s + 4·13-s + 35·16-s + 2·17-s − 4·19-s − 20·20-s − 8·22-s + 8·23-s + 4·25-s − 16·26-s + 10·29-s − 8·31-s − 56·32-s − 8·34-s + 8·37-s + 16·38-s + 40·40-s + 6·41-s − 10·43-s + 20·44-s − 32·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 0.894·5-s − 7.07·8-s + 2.52·10-s + 0.603·11-s + 1.10·13-s + 35/4·16-s + 0.485·17-s − 0.917·19-s − 4.47·20-s − 1.70·22-s + 1.66·23-s + 4/5·25-s − 3.13·26-s + 1.85·29-s − 1.43·31-s − 9.89·32-s − 1.37·34-s + 1.31·37-s + 2.59·38-s + 6.32·40-s + 0.937·41-s − 1.52·43-s + 3.01·44-s − 4.71·46-s + 1.75·47-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
| \(L(1)\) |
\(\approx\) |
\(0.8125222686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8125222686\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 12 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 7 T^{2} + 12 T^{3} + 152 T^{4} + 12 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 30 T^{2} + 96 T^{3} - 845 T^{4} + 96 p T^{5} + 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 10 T + 24 T^{2} - 180 T^{3} + 2035 T^{4} - 180 p T^{5} + 24 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 324 T^{3} - 2373 T^{4} + 324 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 22 T + 232 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 26 T + 304 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 34 T^{2} - 4173 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 54 T^{2} + 576 T^{3} + 12667 T^{4} + 576 p T^{5} + 54 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 324 p T^{5} - 88 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T - 55 T^{2} - 618 T^{3} - 4620 T^{4} - 618 p T^{5} - 55 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.17229311293721084919335946962, −7.08123204245869316110482671890, −6.86375258375659370181209398961, −6.41645055344304494215405708390, −6.29825164277218515394472840297, −6.01002937547767615924401703347, −5.89285723440028100597334001316, −5.60563578632556473619879848351, −5.56229835320021658678277665493, −4.86349834112307851939846972420, −4.63131605175908964682872831481, −4.55322684156876728619436642870, −4.26750867928268588966002406050, −3.85458270703375174296988063953, −3.56817142489272876417774876942, −3.32019966378888857729744195819, −3.04092346490339560348835931315, −2.76740580576426908453101366868, −2.65029561954068070813646303162, −2.15518873827834609591791834492, −1.68557795047085883458816270455, −1.40616753698785388902155265764, −1.26686415028746954820930116703, −0.57760586381494550387557234232, −0.56833349665882021691322437837,
0.56833349665882021691322437837, 0.57760586381494550387557234232, 1.26686415028746954820930116703, 1.40616753698785388902155265764, 1.68557795047085883458816270455, 2.15518873827834609591791834492, 2.65029561954068070813646303162, 2.76740580576426908453101366868, 3.04092346490339560348835931315, 3.32019966378888857729744195819, 3.56817142489272876417774876942, 3.85458270703375174296988063953, 4.26750867928268588966002406050, 4.55322684156876728619436642870, 4.63131605175908964682872831481, 4.86349834112307851939846972420, 5.56229835320021658678277665493, 5.60563578632556473619879848351, 5.89285723440028100597334001316, 6.01002937547767615924401703347, 6.29825164277218515394472840297, 6.41645055344304494215405708390, 6.86375258375659370181209398961, 7.08123204245869316110482671890, 7.17229311293721084919335946962
