| L(s) = 1 | − 2·2-s + 4·3-s − 8·6-s + 2·8-s + 10·9-s + 8·11-s + 6·13-s − 16-s + 2·17-s − 20·18-s + 2·19-s − 16·22-s − 8·23-s + 8·24-s − 12·26-s + 20·27-s + 20·29-s − 4·31-s + 32·33-s − 4·34-s − 4·38-s + 24·39-s + 20·41-s + 8·43-s + 16·46-s − 4·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s − 3.26·6-s + 0.707·8-s + 10/3·9-s + 2.41·11-s + 1.66·13-s − 1/4·16-s + 0.485·17-s − 4.71·18-s + 0.458·19-s − 3.41·22-s − 1.66·23-s + 1.63·24-s − 2.35·26-s + 3.84·27-s + 3.71·29-s − 0.718·31-s + 5.57·33-s − 0.685·34-s − 0.648·38-s + 3.84·39-s + 3.12·41-s + 1.21·43-s + 2.35·46-s − 0.583·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.05395889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.05395889\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 9 T^{4} + 3 p^{2} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 p T^{2} + 16 T^{3} + 103 T^{4} + 16 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 56 T^{2} - 256 T^{3} + 978 T^{4} - 256 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6 T + 60 T^{2} - 232 T^{3} + 1219 T^{4} - 232 p T^{5} + 60 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 28 T^{2} + 40 T^{3} + 251 T^{4} + 40 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 2 T + 36 T^{2} - 104 T^{3} + 687 T^{4} - 104 p T^{5} + 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 8 T + 62 T^{2} + 336 T^{3} + 1879 T^{4} + 336 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} + 236 T^{3} + 1370 T^{4} + 236 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 96 T^{2} - 8 T^{3} + 4930 T^{4} - 8 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 20 T + 302 T^{2} - 2832 T^{3} + 21691 T^{4} - 2832 p T^{5} + 302 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 76 T^{2} - 328 T^{3} + 3414 T^{4} - 328 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 196 T^{2} + 1392 T^{3} + 15451 T^{4} + 1392 p T^{5} + 196 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 10 T + 244 T^{2} - 1672 T^{3} + 21815 T^{4} - 1672 p T^{5} + 244 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 20 T + 210 T^{2} + 1296 T^{3} + 8331 T^{4} + 1296 p T^{5} + 210 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 180 T^{2} - 224 T^{3} + 15386 T^{4} - 224 p T^{5} + 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 18 T + 188 T^{2} - 1144 T^{3} + 8583 T^{4} - 1144 p T^{5} + 188 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4 T + 28 T^{2} - 308 T^{3} + 10970 T^{4} - 308 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 312 T^{2} - 3716 T^{3} + 39426 T^{4} - 3716 p T^{5} + 312 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 28 T + 476 T^{2} - 5388 T^{3} + 53158 T^{4} - 5388 p T^{5} + 476 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 68 T^{2} + 368 T^{3} + 4906 T^{4} + 368 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3:S_4$ | \( 1 - 20 T + 404 T^{2} - 5164 T^{3} + 58358 T^{4} - 5164 p T^{5} + 404 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.19486724658767724019178576193, −6.11490331080120859838373806498, −5.84661034385972248514016819332, −5.44041643848483301544225088598, −5.20518850101923157110084351111, −4.97174111206264922356876369587, −4.67380939803969290220056052118, −4.44818827577493141292306291539, −4.40855974215904911686868450628, −4.13217414832931042431673894736, −3.96170303124733753666106048358, −3.68922822501375424007833532641, −3.54343412101246464047359541165, −3.37810780552581810998870078853, −3.20234684809325395012736556619, −2.84928132387366183868855663046, −2.70614106341405747399478472611, −2.36040449207152085604723027132, −1.94803768256892191332847401833, −1.90959044423804297946054798185, −1.63247060116833111382089343012, −1.17993749177348907224941681853, −0.940137677838597890420795055465, −0.886178972007289519101888924574, −0.53547679194959514811100462074,
0.53547679194959514811100462074, 0.886178972007289519101888924574, 0.940137677838597890420795055465, 1.17993749177348907224941681853, 1.63247060116833111382089343012, 1.90959044423804297946054798185, 1.94803768256892191332847401833, 2.36040449207152085604723027132, 2.70614106341405747399478472611, 2.84928132387366183868855663046, 3.20234684809325395012736556619, 3.37810780552581810998870078853, 3.54343412101246464047359541165, 3.68922822501375424007833532641, 3.96170303124733753666106048358, 4.13217414832931042431673894736, 4.40855974215904911686868450628, 4.44818827577493141292306291539, 4.67380939803969290220056052118, 4.97174111206264922356876369587, 5.20518850101923157110084351111, 5.44041643848483301544225088598, 5.84661034385972248514016819332, 6.11490331080120859838373806498, 6.19486724658767724019178576193
