| L(s) = 1 | − 4·5-s + 12·7-s + 4·11-s − 4·17-s + 36·19-s − 56·23-s − 22·25-s − 48·35-s − 100·43-s − 188·47-s − 142·49-s − 16·55-s − 180·61-s − 356·73-s + 48·77-s + 136·83-s + 16·85-s − 144·95-s + 320·101-s + 224·115-s − 48·119-s − 178·121-s − 200·125-s + 127-s + 131-s + 432·133-s + 137-s + ⋯ |
| L(s) = 1 | − 4/5·5-s + 12/7·7-s + 4/11·11-s − 0.235·17-s + 1.89·19-s − 2.43·23-s − 0.879·25-s − 1.37·35-s − 2.32·43-s − 4·47-s − 2.89·49-s − 0.290·55-s − 2.95·61-s − 4.87·73-s + 0.623·77-s + 1.63·83-s + 0.188·85-s − 1.51·95-s + 3.16·101-s + 1.94·115-s − 0.403·119-s − 1.47·121-s − 8/5·125-s + 0.00787·127-s + 0.00763·131-s + 3.24·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4572455738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4572455738\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 36 T + 272 T^{2} - 1260 T^{3} + 5034 p T^{4} - 1260 p^{2} T^{5} + 272 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 5 | \( ( 1 + 2 T + 17 T^{2} + 186 T^{3} + 404 T^{4} + 186 p^{2} T^{5} + 17 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 6 T + 125 T^{2} - 762 T^{3} + 8208 T^{4} - 762 p^{2} T^{5} + 125 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 95 T^{2} - 786 T^{3} + 22358 T^{4} - 786 p^{2} T^{5} + 95 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 436 T^{2} + 90822 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 2 T + 977 T^{2} + 1194 T^{3} + 397604 T^{4} + 1194 p^{2} T^{5} + 977 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 28 T + 1202 T^{2} + 7836 T^{3} + 409994 T^{4} + 7836 p^{2} T^{5} + 1202 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 - 2804 T^{2} + 4591528 T^{4} - 5482802972 T^{6} + 5092960928014 T^{8} - 5482802972 p^{4} T^{10} + 4591528 p^{8} T^{12} - 2804 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 5072 T^{2} + 12813916 T^{4} - 20816509232 T^{6} + 23644591590214 T^{8} - 20816509232 p^{4} T^{10} + 12813916 p^{8} T^{12} - 5072 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 2500 T^{2} + 3928422 T^{4} - 2500 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 3524 T^{2} + 6992968 T^{4} - 9367576652 T^{6} + 13728001648654 T^{8} - 9367576652 p^{4} T^{10} + 6992968 p^{8} T^{12} - 3524 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 50 T + 127 p T^{2} + 148550 T^{3} + 11741176 T^{4} + 148550 p^{2} T^{5} + 127 p^{5} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 2 p T + 11615 T^{2} + 652878 T^{3} + 41645558 T^{4} + 652878 p^{2} T^{5} + 11615 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 - 2180 T^{2} + 27338824 T^{4} - 48735585740 T^{6} + 307111976245966 T^{8} - 48735585740 p^{4} T^{10} + 27338824 p^{8} T^{12} - 2180 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 5144 T^{2} + 44001148 T^{4} - 148858605992 T^{6} + 762573907571014 T^{8} - 148858605992 p^{4} T^{10} + 44001148 p^{8} T^{12} - 5144 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 90 T + 13109 T^{2} + 989670 T^{3} + 70115496 T^{4} + 989670 p^{2} T^{5} + 13109 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 13472 T^{2} + 56475580 T^{4} - 2272835936 T^{6} - 630219454756922 T^{8} - 2272835936 p^{4} T^{10} + 56475580 p^{8} T^{12} - 13472 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 14744 T^{2} + 102661948 T^{4} - 427102940072 T^{6} + 1684721683685254 T^{8} - 427102940072 p^{4} T^{10} + 102661948 p^{8} T^{12} - 14744 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 178 T + 29317 T^{2} + 2968006 T^{3} + 256241704 T^{4} + 2968006 p^{2} T^{5} + 29317 p^{4} T^{6} + 178 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 + 2728 T^{2} + 145271356 T^{4} + 296261219608 T^{6} + 8275750069351174 T^{8} + 296261219608 p^{4} T^{10} + 145271356 p^{8} T^{12} + 2728 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 - 68 T + 23762 T^{2} - 1145796 T^{3} + 232418954 T^{4} - 1145796 p^{2} T^{5} + 23762 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 - 37028 T^{2} + 668244808 T^{4} - 7948953694316 T^{6} + 71113986292288270 T^{8} - 7948953694316 p^{4} T^{10} + 668244808 p^{8} T^{12} - 37028 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( 1 - 35720 T^{2} + 780372124 T^{4} - 11578494731960 T^{6} + 126138998689341766 T^{8} - 11578494731960 p^{4} T^{10} + 780372124 p^{8} T^{12} - 35720 p^{12} T^{14} + p^{16} T^{16} \) |
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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
−3.38380599347498942512853825756, −3.30202897299886095564526074039, −3.20891902588254496732685627797, −3.05444668596150923816355466758, −3.03179668699804211614165814330, −2.97847969435191256447114639103, −2.84829407021279241899146337964, −2.84409820671548975817261436802, −2.45253119161488293446022522173, −2.28359577113474650741824420035, −2.15710360823120919971395382399, −1.91704320301905132817471131607, −1.83964209562744990597472380666, −1.76916968118187837877023604976, −1.68592966361625162296916359264, −1.61621669151163732167682861734, −1.48824951607936569679773248289, −1.45635668620838897693280777254, −1.21047699156466606806829703097, −1.05066624438220007759474270883, −0.69220142416629661380447346930, −0.54878309901546151222170478053, −0.44240590116589108764396387636, −0.13185267270563884268271335206, −0.096006371250065977008225490367,
0.096006371250065977008225490367, 0.13185267270563884268271335206, 0.44240590116589108764396387636, 0.54878309901546151222170478053, 0.69220142416629661380447346930, 1.05066624438220007759474270883, 1.21047699156466606806829703097, 1.45635668620838897693280777254, 1.48824951607936569679773248289, 1.61621669151163732167682861734, 1.68592966361625162296916359264, 1.76916968118187837877023604976, 1.83964209562744990597472380666, 1.91704320301905132817471131607, 2.15710360823120919971395382399, 2.28359577113474650741824420035, 2.45253119161488293446022522173, 2.84409820671548975817261436802, 2.84829407021279241899146337964, 2.97847969435191256447114639103, 3.03179668699804211614165814330, 3.05444668596150923816355466758, 3.20891902588254496732685627797, 3.30202897299886095564526074039, 3.38380599347498942512853825756
Plot not available for L-functions of degree greater than 10.
