Dirichlet series
| L(s) = 1 | + 5·3-s − 20·7-s + 20·9-s + 2·11-s + 5·13-s − 2·16-s − 10·17-s − 100·21-s + 15·23-s + 65·27-s − 10·29-s + 17·31-s + 10·33-s − 15·37-s + 25·39-s + 12·41-s + 25·47-s − 10·48-s + 140·49-s − 50·51-s + 20·59-s − 23·61-s − 400·63-s + 75·69-s + 22·71-s + 40·73-s − 40·77-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 7.55·7-s + 20/3·9-s + 0.603·11-s + 1.38·13-s − 1/2·16-s − 2.42·17-s − 21.8·21-s + 3.12·23-s + 12.5·27-s − 1.85·29-s + 3.05·31-s + 1.74·33-s − 2.46·37-s + 4.00·39-s + 1.87·41-s + 3.64·47-s − 1.44·48-s + 20·49-s − 7.00·51-s + 2.60·59-s − 2.94·61-s − 50.3·63-s + 9.02·69-s + 2.61·71-s + 4.68·73-s − 4.55·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.48085\times 10^{11}\) |
| Root analytic conductor: | \(2.23397\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{64} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(38.35740388\) |
| \(L(\frac12)\) | \(\approx\) | \(38.35740388\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + p T^{4} + 15 T^{7} + 3 T^{8} + 15 p T^{9} + 15 T^{11} + 17 p T^{12} + 15 p T^{13} + 75 T^{14} + 15 T^{15} + 695 T^{16} + 15 p T^{17} + 75 p^{2} T^{18} + 15 p^{4} T^{19} + 17 p^{5} T^{20} + 15 p^{5} T^{21} + 15 p^{8} T^{23} + 3 p^{8} T^{24} + 15 p^{9} T^{25} + p^{13} T^{28} + p^{16} T^{32} \) |
| 3 | \( 1 - 5 T + 5 T^{2} + 10 T^{3} - 2 p^{2} T^{4} + 5 T^{5} + 10 T^{6} - 55 T^{7} - 67 T^{8} + 485 T^{9} - 295 T^{10} - 685 T^{11} + 1174 T^{12} - 1210 T^{13} + 25 p T^{14} + 85 p T^{15} + 3145 T^{16} + 85 p^{2} T^{17} + 25 p^{3} T^{18} - 1210 p^{3} T^{19} + 1174 p^{4} T^{20} - 685 p^{5} T^{21} - 295 p^{6} T^{22} + 485 p^{7} T^{23} - 67 p^{8} T^{24} - 55 p^{9} T^{25} + 10 p^{10} T^{26} + 5 p^{11} T^{27} - 2 p^{14} T^{28} + 10 p^{13} T^{29} + 5 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( ( 1 + 10 T + 80 T^{2} + 65 p T^{3} + 318 p T^{4} + 9040 T^{5} + 32660 T^{6} + 102420 T^{7} + 289171 T^{8} + 102420 p T^{9} + 32660 p^{2} T^{10} + 9040 p^{3} T^{11} + 318 p^{5} T^{12} + 65 p^{6} T^{13} + 80 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( 1 - 2 T - 14 T^{2} - 25 T^{3} + 200 T^{4} + 294 T^{5} - 1138 T^{6} + 3959 T^{7} + 11175 T^{8} - 44700 T^{9} - 582185 T^{10} + 265145 T^{11} + 575815 p T^{12} + 7200550 T^{13} - 39534450 T^{14} - 86696095 T^{15} + 419723965 T^{16} - 86696095 p T^{17} - 39534450 p^{2} T^{18} + 7200550 p^{3} T^{19} + 575815 p^{5} T^{20} + 265145 p^{5} T^{21} - 582185 p^{6} T^{22} - 44700 p^{7} T^{23} + 11175 p^{8} T^{24} + 3959 p^{9} T^{25} - 1138 p^{10} T^{26} + 294 p^{11} T^{27} + 200 p^{12} T^{28} - 25 p^{13} T^{29} - 14 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 5 T - 35 T^{2} + 285 T^{3} + 237 T^{4} - 6810 T^{5} + 630 p T^{6} + 88350 T^{7} - 205077 T^{8} - 66480 p T^{9} + 2394165 T^{10} + 12416265 T^{11} - 3034992 p T^{12} - 192488610 T^{13} + 991155900 T^{14} + 1243358605 T^{15} - 16828784405 T^{16} + 1243358605 p T^{17} + 991155900 p^{2} T^{18} - 192488610 p^{3} T^{19} - 3034992 p^{5} T^{20} + 12416265 p^{5} T^{21} + 2394165 p^{6} T^{22} - 66480 p^{8} T^{23} - 205077 p^{8} T^{24} + 88350 p^{9} T^{25} + 630 p^{11} T^{26} - 6810 p^{11} T^{27} + 237 p^{12} T^{28} + 285 p^{13} T^{29} - 35 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 10 T + 20 T^{2} + 155 T^{3} + 2447 T^{4} + 8250 T^{5} + 14695 T^{6} + 267590 T^{7} + 1341228 T^{8} + 1528695 T^{9} + 16317320 T^{10} + 131975210 T^{11} + 197988339 T^{12} + 457713055 T^{13} + 8949849450 T^{14} + 25602881915 T^{15} + 3991989945 T^{16} + 25602881915 p T^{17} + 8949849450 p^{2} T^{18} + 457713055 p^{3} T^{19} + 197988339 p^{4} T^{20} + 131975210 p^{5} T^{21} + 16317320 p^{6} T^{22} + 1528695 p^{7} T^{23} + 1341228 p^{8} T^{24} + 267590 p^{9} T^{25} + 14695 p^{10} T^{26} + 8250 p^{11} T^{27} + 2447 p^{12} T^{28} + 155 p^{13} T^{29} + 20 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 56 T^{2} + 105 T^{3} + 1435 T^{4} - 2190 T^{5} - 20245 T^{6} - 64785 T^{7} + 361010 T^{8} + 3137895 T^{9} - 6853328 T^{10} - 57545925 T^{11} - 35850907 T^{12} + 763773615 T^{13} + 4404873155 T^{14} - 5824389045 T^{15} - 105689955635 T^{16} - 5824389045 p T^{17} + 4404873155 p^{2} T^{18} + 763773615 p^{3} T^{19} - 35850907 p^{4} T^{20} - 57545925 p^{5} T^{21} - 6853328 p^{6} T^{22} + 3137895 p^{7} T^{23} + 361010 p^{8} T^{24} - 64785 p^{9} T^{25} - 20245 p^{10} T^{26} - 2190 p^{11} T^{27} + 1435 p^{12} T^{28} + 105 p^{13} T^{29} - 56 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 15 T + 40 T^{2} + 825 T^{3} - 8378 T^{4} + 13260 T^{5} + 283595 T^{6} - 2110140 T^{7} + 1610633 T^{8} + 2430255 p T^{9} - 315082910 T^{10} + 92278560 T^{11} + 6702504734 T^{12} - 33091910460 T^{13} + 1308037450 p T^{14} + 465716917005 T^{15} - 3302717826005 T^{16} + 465716917005 p T^{17} + 1308037450 p^{3} T^{18} - 33091910460 p^{3} T^{19} + 6702504734 p^{4} T^{20} + 92278560 p^{5} T^{21} - 315082910 p^{6} T^{22} + 2430255 p^{8} T^{23} + 1610633 p^{8} T^{24} - 2110140 p^{9} T^{25} + 283595 p^{10} T^{26} + 13260 p^{11} T^{27} - 8378 p^{12} T^{28} + 825 p^{13} T^{29} + 40 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 10 T - 41 T^{2} - 715 T^{3} - 310 T^{4} + 11145 T^{5} - 5815 T^{6} + 552005 T^{7} + 4209765 T^{8} - 28718835 T^{9} - 192187898 T^{10} + 497631020 T^{11} + 1710902153 T^{12} - 2542283645 T^{13} + 143637875895 T^{14} - 7028169115 T^{15} - 6699925029995 T^{16} - 7028169115 p T^{17} + 143637875895 p^{2} T^{18} - 2542283645 p^{3} T^{19} + 1710902153 p^{4} T^{20} + 497631020 p^{5} T^{21} - 192187898 p^{6} T^{22} - 28718835 p^{7} T^{23} + 4209765 p^{8} T^{24} + 552005 p^{9} T^{25} - 5815 p^{10} T^{26} + 11145 p^{11} T^{27} - 310 p^{12} T^{28} - 715 p^{13} T^{29} - 41 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 17 T + 76 T^{2} + 90 T^{3} - 1140 T^{4} + 7899 T^{5} - 8073 T^{6} - 1006311 T^{7} + 7438350 T^{8} - 20749125 T^{9} + 82197165 T^{10} - 48235455 T^{11} - 2486498910 T^{12} - 2243984925 T^{13} + 116700446025 T^{14} - 1015121786420 T^{15} + 7153063167265 T^{16} - 1015121786420 p T^{17} + 116700446025 p^{2} T^{18} - 2243984925 p^{3} T^{19} - 2486498910 p^{4} T^{20} - 48235455 p^{5} T^{21} + 82197165 p^{6} T^{22} - 20749125 p^{7} T^{23} + 7438350 p^{8} T^{24} - 1006311 p^{9} T^{25} - 8073 p^{10} T^{26} + 7899 p^{11} T^{27} - 1140 p^{12} T^{28} + 90 p^{13} T^{29} + 76 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 15 T - 65 T^{2} - 2120 T^{3} - 5283 T^{4} + 102770 T^{5} + 704685 T^{6} - 874895 T^{7} - 30470582 T^{8} - 113737330 T^{9} + 713157310 T^{10} + 7276042070 T^{11} - 4824012096 T^{12} - 300677700795 T^{13} - 997703881150 T^{14} + 5459180729565 T^{15} + 63764728062245 T^{16} + 5459180729565 p T^{17} - 997703881150 p^{2} T^{18} - 300677700795 p^{3} T^{19} - 4824012096 p^{4} T^{20} + 7276042070 p^{5} T^{21} + 713157310 p^{6} T^{22} - 113737330 p^{7} T^{23} - 30470582 p^{8} T^{24} - 874895 p^{9} T^{25} + 704685 p^{10} T^{26} + 102770 p^{11} T^{27} - 5283 p^{12} T^{28} - 2120 p^{13} T^{29} - 65 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 12 T - 9 T^{2} + 750 T^{3} - 3325 T^{4} + 12744 T^{5} - 158928 T^{6} - 631671 T^{7} + 20656425 T^{8} - 90989700 T^{9} - 257475435 T^{10} + 3087924495 T^{11} - 16324025585 T^{12} + 144568829925 T^{13} - 529256276700 T^{14} - 5801689483845 T^{15} + 73492808687165 T^{16} - 5801689483845 p T^{17} - 529256276700 p^{2} T^{18} + 144568829925 p^{3} T^{19} - 16324025585 p^{4} T^{20} + 3087924495 p^{5} T^{21} - 257475435 p^{6} T^{22} - 90989700 p^{7} T^{23} + 20656425 p^{8} T^{24} - 631671 p^{9} T^{25} - 158928 p^{10} T^{26} + 12744 p^{11} T^{27} - 3325 p^{12} T^{28} + 750 p^{13} T^{29} - 9 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 245 T^{2} - 180 T^{3} + 28176 T^{4} - 33015 T^{5} + 2039890 T^{6} - 2606415 T^{7} + 103354571 T^{8} - 2606415 p T^{9} + 2039890 p^{2} T^{10} - 33015 p^{3} T^{11} + 28176 p^{4} T^{12} - 180 p^{5} T^{13} + 245 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 25 T + 185 T^{2} + 670 T^{3} - 18298 T^{4} + 93435 T^{5} + 130060 T^{6} - 4760990 T^{7} + 39616338 T^{8} - 97725060 T^{9} - 1590577540 T^{10} + 11407491115 T^{11} + 54769909449 T^{12} - 815910097870 T^{13} + 1369781617800 T^{14} + 20957288056015 T^{15} - 195819882577005 T^{16} + 20957288056015 p T^{17} + 1369781617800 p^{2} T^{18} - 815910097870 p^{3} T^{19} + 54769909449 p^{4} T^{20} + 11407491115 p^{5} T^{21} - 1590577540 p^{6} T^{22} - 97725060 p^{7} T^{23} + 39616338 p^{8} T^{24} - 4760990 p^{9} T^{25} + 130060 p^{10} T^{26} + 93435 p^{11} T^{27} - 18298 p^{12} T^{28} + 670 p^{13} T^{29} + 185 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 195 T^{2} + 345 T^{3} + 19967 T^{4} - 101775 T^{5} - 1478310 T^{6} + 12713205 T^{7} + 82463973 T^{8} - 1005197295 T^{9} - 2411798670 T^{10} + 21372495 p^{2} T^{11} - 65848057976 T^{12} - 2726767689810 T^{13} + 12151368928350 T^{14} + 59069038980705 T^{15} - 799918596084655 T^{16} + 59069038980705 p T^{17} + 12151368928350 p^{2} T^{18} - 2726767689810 p^{3} T^{19} - 65848057976 p^{4} T^{20} + 21372495 p^{7} T^{21} - 2411798670 p^{6} T^{22} - 1005197295 p^{7} T^{23} + 82463973 p^{8} T^{24} + 12713205 p^{9} T^{25} - 1478310 p^{10} T^{26} - 101775 p^{11} T^{27} + 19967 p^{12} T^{28} + 345 p^{13} T^{29} - 195 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 20 T - 11 T^{2} + 2450 T^{3} - 1585 T^{4} - 276750 T^{5} + 722660 T^{6} + 24764150 T^{7} - 125600535 T^{8} - 1583285925 T^{9} + 11700534202 T^{10} + 95855471960 T^{11} - 1043353559512 T^{12} - 4055048016125 T^{13} + 76687974499320 T^{14} + 74363062087400 T^{15} - 4676890271000195 T^{16} + 74363062087400 p T^{17} + 76687974499320 p^{2} T^{18} - 4055048016125 p^{3} T^{19} - 1043353559512 p^{4} T^{20} + 95855471960 p^{5} T^{21} + 11700534202 p^{6} T^{22} - 1583285925 p^{7} T^{23} - 125600535 p^{8} T^{24} + 24764150 p^{9} T^{25} + 722660 p^{10} T^{26} - 276750 p^{11} T^{27} - 1585 p^{12} T^{28} + 2450 p^{13} T^{29} - 11 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 23 T + 201 T^{2} + 855 T^{3} + 9120 T^{4} + 134479 T^{5} + 582662 T^{6} - 3462591 T^{7} - 21827775 T^{8} + 110229025 T^{9} - 1046667285 T^{10} - 26820569905 T^{11} - 67038400560 T^{12} + 959914543800 T^{13} + 8373562323325 T^{14} + 65588615075580 T^{15} + 621552345011915 T^{16} + 65588615075580 p T^{17} + 8373562323325 p^{2} T^{18} + 959914543800 p^{3} T^{19} - 67038400560 p^{4} T^{20} - 26820569905 p^{5} T^{21} - 1046667285 p^{6} T^{22} + 110229025 p^{7} T^{23} - 21827775 p^{8} T^{24} - 3462591 p^{9} T^{25} + 582662 p^{10} T^{26} + 134479 p^{11} T^{27} + 9120 p^{12} T^{28} + 855 p^{13} T^{29} + 201 p^{14} T^{30} + 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 60 T^{2} + 1055 T^{3} + 5532 T^{4} - 27790 T^{5} + 435255 T^{6} + 5776185 T^{7} + 18151918 T^{8} + 177071270 T^{9} + 1654147440 T^{10} + 31995035340 T^{11} + 216562649614 T^{12} + 565177305 T^{13} + 20520454745800 T^{14} + 153888332201315 T^{15} - 83982510456855 T^{16} + 153888332201315 p T^{17} + 20520454745800 p^{2} T^{18} + 565177305 p^{3} T^{19} + 216562649614 p^{4} T^{20} + 31995035340 p^{5} T^{21} + 1654147440 p^{6} T^{22} + 177071270 p^{7} T^{23} + 18151918 p^{8} T^{24} + 5776185 p^{9} T^{25} + 435255 p^{10} T^{26} - 27790 p^{11} T^{27} + 5532 p^{12} T^{28} + 1055 p^{13} T^{29} - 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 22 T + 86 T^{2} + 1555 T^{3} - 9355 T^{4} - 76146 T^{5} - 166268 T^{6} + 13485574 T^{7} - 14360775 T^{8} - 839539350 T^{9} + 779008340 T^{10} + 28520365045 T^{11} + 277936351590 T^{12} - 2408579368075 T^{13} - 18701952609675 T^{14} + 63259978277455 T^{15} + 1372736912431065 T^{16} + 63259978277455 p T^{17} - 18701952609675 p^{2} T^{18} - 2408579368075 p^{3} T^{19} + 277936351590 p^{4} T^{20} + 28520365045 p^{5} T^{21} + 779008340 p^{6} T^{22} - 839539350 p^{7} T^{23} - 14360775 p^{8} T^{24} + 13485574 p^{9} T^{25} - 166268 p^{10} T^{26} - 76146 p^{11} T^{27} - 9355 p^{12} T^{28} + 1555 p^{13} T^{29} + 86 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 40 T + 795 T^{2} - 11205 T^{3} + 142062 T^{4} - 1792295 T^{5} + 20782100 T^{6} - 212208105 T^{7} + 2047189218 T^{8} - 20017585070 T^{9} + 191948281620 T^{10} - 1728474118465 T^{11} + 15310683788334 T^{12} - 139966609653660 T^{13} + 1269169022276125 T^{14} - 10944032048556720 T^{15} + 92635248745911395 T^{16} - 10944032048556720 p T^{17} + 1269169022276125 p^{2} T^{18} - 139966609653660 p^{3} T^{19} + 15310683788334 p^{4} T^{20} - 1728474118465 p^{5} T^{21} + 191948281620 p^{6} T^{22} - 20017585070 p^{7} T^{23} + 2047189218 p^{8} T^{24} - 212208105 p^{9} T^{25} + 20782100 p^{10} T^{26} - 1792295 p^{11} T^{27} + 142062 p^{12} T^{28} - 11205 p^{13} T^{29} + 795 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 75 T + 2479 T^{2} - 47495 T^{3} + 595785 T^{4} - 5590990 T^{5} + 49066230 T^{6} - 470839160 T^{7} + 4637260285 T^{8} - 46927296130 T^{9} + 507032254297 T^{10} - 5300851535725 T^{11} + 51719664481188 T^{12} - 499615995672210 T^{13} + 4590535334577830 T^{14} - 37525646623667595 T^{15} + 308797800467018165 T^{16} - 37525646623667595 p T^{17} + 4590535334577830 p^{2} T^{18} - 499615995672210 p^{3} T^{19} + 51719664481188 p^{4} T^{20} - 5300851535725 p^{5} T^{21} + 507032254297 p^{6} T^{22} - 46927296130 p^{7} T^{23} + 4637260285 p^{8} T^{24} - 470839160 p^{9} T^{25} + 49066230 p^{10} T^{26} - 5590990 p^{11} T^{27} + 595785 p^{12} T^{28} - 47495 p^{13} T^{29} + 2479 p^{14} T^{30} - 75 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 25 T + 85 T^{2} + 3445 T^{3} - 48763 T^{4} + 117570 T^{5} + 4696850 T^{6} - 66104660 T^{7} + 125540013 T^{8} + 5547018030 T^{9} - 65280110165 T^{10} + 168810261805 T^{11} + 4350665117864 T^{12} - 61076064667510 T^{13} + 209007822945750 T^{14} + 2907275202214705 T^{15} - 45160254209839055 T^{16} + 2907275202214705 p T^{17} + 209007822945750 p^{2} T^{18} - 61076064667510 p^{3} T^{19} + 4350665117864 p^{4} T^{20} + 168810261805 p^{5} T^{21} - 65280110165 p^{6} T^{22} + 5547018030 p^{7} T^{23} + 125540013 p^{8} T^{24} - 66104660 p^{9} T^{25} + 4696850 p^{10} T^{26} + 117570 p^{11} T^{27} - 48763 p^{12} T^{28} + 3445 p^{13} T^{29} + 85 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 5 T + 4 T^{2} - 1630 T^{3} + 17600 T^{4} - 56085 T^{5} + 1656845 T^{6} - 16334215 T^{7} + 86041950 T^{8} - 1321600995 T^{9} + 6089922937 T^{10} - 18878919385 T^{11} + 744947140478 T^{12} - 111112025765 T^{13} - 57628410305175 T^{14} + 125077436483270 T^{15} + 221939614719685 T^{16} + 125077436483270 p T^{17} - 57628410305175 p^{2} T^{18} - 111112025765 p^{3} T^{19} + 744947140478 p^{4} T^{20} - 18878919385 p^{5} T^{21} + 6089922937 p^{6} T^{22} - 1321600995 p^{7} T^{23} + 86041950 p^{8} T^{24} - 16334215 p^{9} T^{25} + 1656845 p^{10} T^{26} - 56085 p^{11} T^{27} + 17600 p^{12} T^{28} - 1630 p^{13} T^{29} + 4 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 40 T + 805 T^{2} - 11110 T^{3} + 139027 T^{4} - 1371740 T^{5} + 4662030 T^{6} + 123636885 T^{7} - 2784947487 T^{8} + 41739944450 T^{9} - 523970765595 T^{10} + 4740615289385 T^{11} - 21492045511501 T^{12} - 79629050843495 T^{13} + 3491991526055250 T^{14} - 60854539653934685 T^{15} + 734888534412891145 T^{16} - 60854539653934685 p T^{17} + 3491991526055250 p^{2} T^{18} - 79629050843495 p^{3} T^{19} - 21492045511501 p^{4} T^{20} + 4740615289385 p^{5} T^{21} - 523970765595 p^{6} T^{22} + 41739944450 p^{7} T^{23} - 2784947487 p^{8} T^{24} + 123636885 p^{9} T^{25} + 4662030 p^{10} T^{26} - 1371740 p^{11} T^{27} + 139027 p^{12} T^{28} - 11110 p^{13} T^{29} + 805 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.87691402034306288849831974507, −2.84622465820546800688201692505, −2.78479552189477911817251514721, −2.66040532712428355907613991819, −2.62845431821111200384828755758, −2.40763538325450860008928193306, −2.29979183367674778215056605138, −2.21756852317077890486040940028, −2.17501783487276168944968497319, −2.11676736993369166061875678156, −2.06970167329321192316007237270, −2.06956332681348278142918572623, −1.90580046208649946605001678315, −1.84567072018475272099114708683, −1.61551425853272049366990283496, −1.52574057981633482419162874480, −1.46122195823279898194335520878, −1.25434578451315930045501411256, −0.938101299890577839385381059312, −0.77863013848943149582561579523, −0.76145319698872502959377538504, −0.68700345732693732141276609237, −0.60562834112357745236909079259, −0.52374286363108390194574204620, −0.39055583812199351510146633567, 0.39055583812199351510146633567, 0.52374286363108390194574204620, 0.60562834112357745236909079259, 0.68700345732693732141276609237, 0.76145319698872502959377538504, 0.77863013848943149582561579523, 0.938101299890577839385381059312, 1.25434578451315930045501411256, 1.46122195823279898194335520878, 1.52574057981633482419162874480, 1.61551425853272049366990283496, 1.84567072018475272099114708683, 1.90580046208649946605001678315, 2.06956332681348278142918572623, 2.06970167329321192316007237270, 2.11676736993369166061875678156, 2.17501783487276168944968497319, 2.21756852317077890486040940028, 2.29979183367674778215056605138, 2.40763538325450860008928193306, 2.62845431821111200384828755758, 2.66040532712428355907613991819, 2.78479552189477911817251514721, 2.84622465820546800688201692505, 2.87691402034306288849831974507
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.