| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5·5-s − 2·6-s − 5·7-s + 6·8-s − 2·9-s + 10·10-s + 4·11-s + 2·12-s − 10·13-s + 10·14-s − 5·15-s − 11·16-s − 4·17-s + 4·18-s − 3·19-s − 10·20-s − 5·21-s − 8·22-s + 16·23-s + 6·24-s + 10·25-s + 20·26-s + 3·27-s − 10·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 2.23·5-s − 0.816·6-s − 1.88·7-s + 2.12·8-s − 2/3·9-s + 3.16·10-s + 1.20·11-s + 0.577·12-s − 2.77·13-s + 2.67·14-s − 1.29·15-s − 2.75·16-s − 0.970·17-s + 0.942·18-s − 0.688·19-s − 2.23·20-s − 1.09·21-s − 1.70·22-s + 3.33·23-s + 1.22·24-s + 2·25-s + 3.92·26-s + 0.577·27-s − 1.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2272616660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2272616660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 55 p T^{5} + 18 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( 1 + p T + p T^{2} - 3 p T^{3} - 17 T^{4} - 3 p^{3} T^{5} - T^{6} + 47 T^{7} + 103 T^{8} + 47 p T^{9} - p^{2} T^{10} - 3 p^{6} T^{11} - 17 p^{4} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - T + p T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 2 p T^{6} + 56 T^{7} - 137 T^{8} + 56 p T^{9} + 2 p^{3} T^{10} + 7 p^{3} T^{11} + 8 p^{4} T^{12} - 8 p^{5} T^{13} + p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2}( 1 - p T + 2 p T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 + 5 T + 27 T^{2} + 115 T^{3} + 584 T^{4} + 115 p T^{5} + 27 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} - 120 T^{3} - 740 T^{4} - 3144 T^{5} - 649 T^{6} + 49990 T^{7} + 275299 T^{8} + 49990 p T^{9} - 649 p^{2} T^{10} - 3144 p^{3} T^{11} - 740 p^{4} T^{12} - 120 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 3 T - 26 T^{2} - 21 T^{3} + 252 T^{4} - 1302 T^{5} - 1792 T^{6} + 24084 T^{7} + 90161 T^{8} + 24084 p T^{9} - 1792 p^{2} T^{10} - 1302 p^{3} T^{11} + 252 p^{4} T^{12} - 21 p^{5} T^{13} - 26 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 88 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 12 T + 25 T^{2} + 288 T^{3} - 2471 T^{4} + 288 p T^{5} + 25 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 19 T + 210 T^{2} - 1691 T^{3} + 10649 T^{4} - 1691 p T^{5} + 210 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )( 1 + 11 T + 60 T^{2} - 11 T^{3} - 991 T^{4} - 11 p T^{5} + 60 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 37 | \( 1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 1263548 p T^{9} - 63616 p^{2} T^{10} - 46406 p^{3} T^{11} - 4102 p^{4} T^{12} + 71 p^{5} T^{13} + 112 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - T - 25 T^{2} + 221 T^{3} + 1064 T^{4} + 221 p T^{5} - 25 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 6 T + 7 T^{2} + 690 T^{3} - 6420 T^{4} + 23196 T^{5} + 57521 T^{6} - 1846680 T^{7} + 13766039 T^{8} - 1846680 p T^{9} + 57521 p^{2} T^{10} + 23196 p^{3} T^{11} - 6420 p^{4} T^{12} + 690 p^{5} T^{13} + 7 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 3945120 p T^{9} - 99751 p^{2} T^{10} - 102312 p^{3} T^{11} - 9840 p^{4} T^{12} - 420 p^{5} T^{13} + 103 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 18 T + 239 T^{2} + 1374 T^{3} + 5292 T^{4} - 7812 T^{5} + 7967 p T^{6} + 8281674 T^{7} + 95738891 T^{8} + 8281674 p T^{9} + 7967 p^{3} T^{10} - 7812 p^{3} T^{11} + 5292 p^{4} T^{12} + 1374 p^{5} T^{13} + 239 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 18 T + 141 T^{2} + 538 T^{3} - 19524 T^{4} + 199396 T^{5} - 460901 T^{6} - 9059916 T^{7} + 127866487 T^{8} - 9059916 p T^{9} - 460901 p^{2} T^{10} + 199396 p^{3} T^{11} - 19524 p^{4} T^{12} + 538 p^{5} T^{13} + 141 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 1691 p T^{5} + 138 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 15 T + 208 T^{2} - 2445 T^{3} + 29070 T^{4} - 272130 T^{5} + 2789288 T^{6} - 321120 p T^{7} + 210261239 T^{8} - 321120 p^{2} T^{9} + 2789288 p^{2} T^{10} - 272130 p^{3} T^{11} + 29070 p^{4} T^{12} - 2445 p^{5} T^{13} + 208 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 9 T - 11 T^{2} + 1446 T^{3} - 17334 T^{4} + 85743 T^{5} + 200726 T^{6} - 10774872 T^{7} + 128238887 T^{8} - 10774872 p T^{9} + 200726 p^{2} T^{10} + 85743 p^{3} T^{11} - 17334 p^{4} T^{12} + 1446 p^{5} T^{13} - 11 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 765 p T^{5} - 2 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 192 p T^{5} - 50 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 7 T - 63 T^{2} + 185 T^{3} + 11276 T^{4} + 185 p T^{5} - 63 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24913816200202102064102658967, −6.67685639934637494595092862471, −6.55355054841958374355120279511, −6.54569970800395499160474917463, −6.53527789999421348087676857840, −6.51236454410943752405281615498, −6.05871841165443151200266022321, −5.63087306893822093115757178744, −5.46576790372544036688142814244, −5.16924825591601117619993026763, −4.87002755717468373879739907980, −4.65578688271586897474356695260, −4.64750587798775663100186465814, −4.55355169743199541460354082713, −4.36148559571460170562599272434, −4.13766629233040797137935539461, −3.70075312541349697491267104976, −3.61803503097221804920034954135, −3.18355983905936257973913749824, −2.88548250612170230312294576514, −2.70417713291401016420576223776, −2.60552208919929448933064655200, −2.48967034069081766862086709704, −1.26071485104049961258309706927, −1.12364637577154910260252882813,
1.12364637577154910260252882813, 1.26071485104049961258309706927, 2.48967034069081766862086709704, 2.60552208919929448933064655200, 2.70417713291401016420576223776, 2.88548250612170230312294576514, 3.18355983905936257973913749824, 3.61803503097221804920034954135, 3.70075312541349697491267104976, 4.13766629233040797137935539461, 4.36148559571460170562599272434, 4.55355169743199541460354082713, 4.64750587798775663100186465814, 4.65578688271586897474356695260, 4.87002755717468373879739907980, 5.16924825591601117619993026763, 5.46576790372544036688142814244, 5.63087306893822093115757178744, 6.05871841165443151200266022321, 6.51236454410943752405281615498, 6.53527789999421348087676857840, 6.54569970800395499160474917463, 6.55355054841958374355120279511, 6.67685639934637494595092862471, 7.24913816200202102064102658967
Plot not available for L-functions of degree greater than 10.
