| L(s) = 1 | + 2·4-s − 18·5-s + 2·7-s + 6·9-s + 18·11-s − 10·13-s − 40·19-s − 36·20-s + 18·23-s + 139·25-s + 4·28-s + 18·29-s + 38·31-s − 36·35-s + 12·36-s + 128·37-s − 126·41-s − 46·43-s + 36·44-s − 108·45-s + 54·47-s + 45·49-s − 20·52-s − 324·55-s + 126·59-s + 62·61-s + 12·63-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.59·5-s + 2/7·7-s + 2/3·9-s + 1.63·11-s − 0.769·13-s − 2.10·19-s − 9/5·20-s + 0.782·23-s + 5.55·25-s + 1/7·28-s + 0.620·29-s + 1.22·31-s − 1.02·35-s + 1/3·36-s + 3.45·37-s − 3.07·41-s − 1.06·43-s + 9/11·44-s − 2.39·45-s + 1.14·47-s + 0.918·49-s − 0.384·52-s − 5.89·55-s + 2.13·59-s + 1.01·61-s + 4/21·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3885541672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3885541672\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 p T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 41 T^{2} + 106 T^{3} - 572 T^{4} + 106 p^{2} T^{5} - 41 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 4518 p^{2} T^{5} + 359 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 10 T - 47 T^{2} - 1910 T^{3} - 23852 T^{4} - 1910 p^{2} T^{5} - 47 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T + 606 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 18 T + 1175 T^{2} - 19206 T^{3} + 915780 T^{4} - 19206 p^{2} T^{5} + 1175 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 29466 p^{2} T^{5} + 1745 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4750 p^{2} T^{5} - 353 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 508662 p^{2} T^{5} + 9329 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 46 T - 1625 T^{2} + 46 p T^{3} + 3604 p^{2} T^{4} + 46 p^{3} T^{5} - 1625 p^{4} T^{6} + 46 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 54 T + 4751 T^{2} - 204066 T^{3} + 11548308 T^{4} - 204066 p^{2} T^{5} + 4751 p^{4} T^{6} - 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T + 10535 T^{2} - 660618 T^{3} + 33793140 T^{4} - 660618 p^{2} T^{5} + 10535 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 60946 p^{2} T^{5} - 2615 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 106 T - 65 T^{2} + 246238 T^{3} + 57123076 T^{4} + 246238 p^{2} T^{5} - 65 p^{4} T^{6} + 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 14 T - 10985 T^{2} + 18214 T^{3} + 84841444 T^{4} + 18214 p^{2} T^{5} - 10985 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−14.54779867993553878089108709069, −13.68761001830931743560191895933, −13.30198515919509624191988871080, −12.77057074571954182052727572579, −12.68396183205092856634498135608, −11.81568293362148607879355783074, −11.77847013109460800075356973662, −11.73242323252372239017402153021, −11.62936365107692201197538088819, −10.89429166603966416421095008930, −10.62887905274075050268212824797, −9.880974393067798906745917632475, −9.812433465428032710850995173630, −8.678128793745155372043192588380, −8.547550303254200435591441805699, −8.333607289168685714296216901136, −7.77789387497478209741735980202, −7.22373635247377566575366078629, −7.01625851001993022562047922659, −6.68753921561256918878904219742, −5.85459802292309601854744857078, −4.44660395771464814953646858087, −4.28355042116739765382981373077, −4.16831666093679399606816968694, −3.08266487875230505300547482247,
3.08266487875230505300547482247, 4.16831666093679399606816968694, 4.28355042116739765382981373077, 4.44660395771464814953646858087, 5.85459802292309601854744857078, 6.68753921561256918878904219742, 7.01625851001993022562047922659, 7.22373635247377566575366078629, 7.77789387497478209741735980202, 8.333607289168685714296216901136, 8.547550303254200435591441805699, 8.678128793745155372043192588380, 9.812433465428032710850995173630, 9.880974393067798906745917632475, 10.62887905274075050268212824797, 10.89429166603966416421095008930, 11.62936365107692201197538088819, 11.73242323252372239017402153021, 11.77847013109460800075356973662, 11.81568293362148607879355783074, 12.68396183205092856634498135608, 12.77057074571954182052727572579, 13.30198515919509624191988871080, 13.68761001830931743560191895933, 14.54779867993553878089108709069
