| L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 4·11-s + 2·13-s − 4·16-s + 4·17-s + 2·19-s − 8·22-s + 8·23-s + 10·25-s + 4·26-s − 6·29-s + 8·31-s − 2·32-s + 8·34-s + 24·37-s + 4·38-s + 20·41-s − 14·43-s − 4·44-s + 16·46-s + 18·47-s − 16·49-s + 20·50-s + 2·52-s − 4·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s − 1.20·11-s + 0.554·13-s − 16-s + 0.970·17-s + 0.458·19-s − 1.70·22-s + 1.66·23-s + 2·25-s + 0.784·26-s − 1.11·29-s + 1.43·31-s − 0.353·32-s + 1.37·34-s + 3.94·37-s + 0.648·38-s + 3.12·41-s − 2.13·43-s − 0.603·44-s + 2.35·46-s + 2.62·47-s − 2.28·49-s + 2.82·50-s + 0.277·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.109338473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.109338473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 46 T^{3} - 212 T^{4} + 46 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 16 T^{2} + 8 T^{3} + 463 T^{4} + 8 p T^{5} - 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 26 T^{2} + 64 T^{3} - 557 T^{4} + 64 p T^{5} + 26 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 90 T^{3} - 36 T^{4} - 90 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 224 T^{2} - 1880 T^{3} + 12895 T^{4} - 1880 p T^{5} + 224 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 67 T^{2} + 14 p T^{3} + 148 p T^{4} + 14 p^{2} T^{5} + 67 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 155 T^{2} - 1350 T^{3} + 11124 T^{4} - 1350 p T^{5} + 155 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 368 T^{3} - 3461 T^{4} - 368 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 82 T^{2} + 80 T^{3} + 5179 T^{4} + 80 p T^{5} - 82 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 71 T^{2} - 90 T^{3} + 5532 T^{4} - 90 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 24 T + 304 T^{2} + 3312 T^{3} + 30903 T^{4} + 3312 p T^{5} + 304 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 10 T - 61 T^{2} + 190 T^{3} + 13780 T^{4} + 190 p T^{5} - 61 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 4 p T^{2} - 3312 T^{3} + 33279 T^{4} - 3312 p T^{5} + 4 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 140 T^{2} - 8 T^{3} + 16831 T^{4} - 8 p T^{5} - 140 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 167 T^{2} + 46 T^{3} + 19444 T^{4} + 46 p T^{5} - 167 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.023850597973445075790888616251, −7.85540533194878010393765174010, −7.74655719710057075433856812629, −7.25660365171826809343233341761, −7.19275361387268676597729975072, −6.86415560382151098919822035226, −6.37519933495994659271715375013, −6.20522323053929088192097169523, −6.12044375209762311012044547932, −5.80484191065818394077837693307, −5.49679997567247016432221331838, −5.28394356656640877628977480755, −4.96917511571914993564749073160, −4.61365356222814938401413189332, −4.54641068576088742078564985340, −4.42931954339714255197124984725, −3.77517802349707257875465153411, −3.66539537263304809469616905532, −3.16443933309406537574191905229, −2.78419605402667272836262289134, −2.69874192466978472295051983661, −2.68581937347491886501962845288, −1.75668513905355026185628795932, −0.957804642011106021142396008386, −0.906463914149593985899368115122,
0.906463914149593985899368115122, 0.957804642011106021142396008386, 1.75668513905355026185628795932, 2.68581937347491886501962845288, 2.69874192466978472295051983661, 2.78419605402667272836262289134, 3.16443933309406537574191905229, 3.66539537263304809469616905532, 3.77517802349707257875465153411, 4.42931954339714255197124984725, 4.54641068576088742078564985340, 4.61365356222814938401413189332, 4.96917511571914993564749073160, 5.28394356656640877628977480755, 5.49679997567247016432221331838, 5.80484191065818394077837693307, 6.12044375209762311012044547932, 6.20522323053929088192097169523, 6.37519933495994659271715375013, 6.86415560382151098919822035226, 7.19275361387268676597729975072, 7.25660365171826809343233341761, 7.74655719710057075433856812629, 7.85540533194878010393765174010, 8.023850597973445075790888616251
