| L(s) = 1 | − 3·3-s + 9·5-s + 7-s − 3·9-s + 36·11-s + 5·13-s − 27·15-s − 2·19-s − 3·21-s − 99·23-s + 22·25-s + 18·27-s − 63·29-s + 7·31-s − 108·33-s + 9·35-s − 64·37-s − 15·39-s − 18·41-s + 46·43-s − 27·45-s + 81·47-s + 24·49-s + 324·55-s + 6·57-s − 126·59-s + 41·61-s + ⋯ |
| L(s) = 1 | − 3-s + 9/5·5-s + 1/7·7-s − 1/3·9-s + 3.27·11-s + 5/13·13-s − 9/5·15-s − 0.105·19-s − 1/7·21-s − 4.30·23-s + 0.879·25-s + 2/3·27-s − 2.17·29-s + 7/31·31-s − 3.27·33-s + 9/35·35-s − 1.72·37-s − 0.384·39-s − 0.439·41-s + 1.06·43-s − 3/5·45-s + 1.72·47-s + 0.489·49-s + 5.89·55-s + 2/19·57-s − 2.13·59-s + 0.672·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.089068161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089068161\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 9 T + 59 T^{2} - 288 T^{3} + 1074 T^{4} - 288 p^{2} T^{5} + 59 p^{4} T^{6} - 9 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 74 T^{3} - 1874 T^{4} + 74 p^{2} T^{5} - 23 p^{4} T^{6} - p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36 T + 683 T^{2} - 9036 T^{3} + 100632 T^{4} - 9036 p^{2} T^{5} + 683 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T - 245 T^{2} + 340 T^{3} + 40114 T^{4} + 340 p^{2} T^{5} - 245 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 648 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 183150 p^{2} T^{5} + 5117 p^{4} T^{6} + 99 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 50400 p^{2} T^{5} + 2123 p^{4} T^{6} + 63 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 7 T - 1217 T^{2} + 4592 T^{3} + 632146 T^{4} + 4592 p^{2} T^{5} - 1217 p^{4} T^{6} - 7 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 32 T + 1806 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 32490 p^{2} T^{5} + 1913 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 23 T - 1320 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 384102 p^{2} T^{5} + 6929 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 844326 p^{2} T^{5} + 11993 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 80156 p^{2} T^{5} + 3787 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 43 T + 10452 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 83 T - 5459 T^{2} - 11122 T^{3} + 70528774 T^{4} - 11122 p^{2} T^{5} - 5459 p^{4} T^{6} + 83 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 81 T + 16289 T^{2} - 1142262 T^{3} + 166474326 T^{4} - 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−9.606747661679046421370270945169, −9.011834624299118780574931708676, −8.933471970740367358970774834565, −8.654328719676497497899593775973, −8.611465574103012347048495433090, −7.83874043363909364404563112541, −7.60465809793590322586196620587, −7.45879734614218366091815576183, −6.97045443707983262170098715538, −6.53934082316317018217289894828, −6.14078357180180067412212514364, −6.13356583486110811311867919421, −6.09318878281470236542862062003, −5.66898602710482737840393133874, −5.53074998367940250915103558501, −4.96542777534912950258757248697, −4.45635333501527627130821515314, −3.94914786953374667369238860954, −3.89939669919011737469950814302, −3.67560794279252365039610890123, −2.87458207401416347660618943505, −1.92157986650593612286107693687, −1.83728588167305081205276489743, −1.72057578928932243733779162326, −0.55238639663421763721218855534,
0.55238639663421763721218855534, 1.72057578928932243733779162326, 1.83728588167305081205276489743, 1.92157986650593612286107693687, 2.87458207401416347660618943505, 3.67560794279252365039610890123, 3.89939669919011737469950814302, 3.94914786953374667369238860954, 4.45635333501527627130821515314, 4.96542777534912950258757248697, 5.53074998367940250915103558501, 5.66898602710482737840393133874, 6.09318878281470236542862062003, 6.13356583486110811311867919421, 6.14078357180180067412212514364, 6.53934082316317018217289894828, 6.97045443707983262170098715538, 7.45879734614218366091815576183, 7.60465809793590322586196620587, 7.83874043363909364404563112541, 8.611465574103012347048495433090, 8.654328719676497497899593775973, 8.933471970740367358970774834565, 9.011834624299118780574931708676, 9.606747661679046421370270945169
