Dirichlet series
| L(s) = 1 | + 16·3-s + 3·5-s + 4·7-s + 136·9-s + 5·11-s + 6·13-s + 48·15-s + 3·17-s + 11·19-s + 64·21-s + 16·23-s − 22·25-s + 816·27-s − 16·29-s + 14·31-s + 80·33-s + 12·35-s + 4·37-s + 96·39-s + 11·41-s + 23·43-s + 408·45-s − 2·47-s − 31·49-s + 48·51-s + 19·53-s + 15·55-s + ⋯ |
| L(s) = 1 | + 9.23·3-s + 1.34·5-s + 1.51·7-s + 45.3·9-s + 1.50·11-s + 1.66·13-s + 12.3·15-s + 0.727·17-s + 2.52·19-s + 13.9·21-s + 3.33·23-s − 4.39·25-s + 157.·27-s − 2.97·29-s + 2.51·31-s + 13.9·33-s + 2.02·35-s + 0.657·37-s + 15.3·39-s + 1.71·41-s + 3.50·43-s + 60.8·45-s − 0.291·47-s − 4.42·49-s + 6.72·51-s + 2.60·53-s + 2.02·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.75080\times 10^{28}\) |
| Root analytic conductor: | \(7.99451\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(850745.1203\) |
| \(L(\frac12)\) | \(\approx\) | \(850745.1203\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{16} \) | |
| 23 | \( ( 1 - T )^{16} \) | |
| 29 | \( ( 1 + T )^{16} \) | |
| good | 5 | \( 1 - 3 T + 31 T^{2} - 19 p T^{3} + 502 T^{4} - 1453 T^{5} + 1096 p T^{6} - 2853 p T^{7} + 43779 T^{8} - 99922 T^{9} + 265306 T^{10} - 525742 T^{11} + 1264458 T^{12} - 2188838 T^{13} + 5201327 T^{14} - 8479994 T^{15} + 23022008 T^{16} - 8479994 p T^{17} + 5201327 p^{2} T^{18} - 2188838 p^{3} T^{19} + 1264458 p^{4} T^{20} - 525742 p^{5} T^{21} + 265306 p^{6} T^{22} - 99922 p^{7} T^{23} + 43779 p^{8} T^{24} - 2853 p^{10} T^{25} + 1096 p^{11} T^{26} - 1453 p^{11} T^{27} + 502 p^{12} T^{28} - 19 p^{14} T^{29} + 31 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 - 4 T + 47 T^{2} - 137 T^{3} + 972 T^{4} - 1999 T^{5} + 12134 T^{6} - 16308 T^{7} + 112920 T^{8} - 92052 T^{9} + 974324 T^{10} - 611743 T^{11} + 8628020 T^{12} - 6093673 T^{13} + 73145687 T^{14} - 56106964 T^{15} + 549676366 T^{16} - 56106964 p T^{17} + 73145687 p^{2} T^{18} - 6093673 p^{3} T^{19} + 8628020 p^{4} T^{20} - 611743 p^{5} T^{21} + 974324 p^{6} T^{22} - 92052 p^{7} T^{23} + 112920 p^{8} T^{24} - 16308 p^{9} T^{25} + 12134 p^{10} T^{26} - 1999 p^{11} T^{27} + 972 p^{12} T^{28} - 137 p^{13} T^{29} + 47 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 5 T + 90 T^{2} - 368 T^{3} + 3750 T^{4} - 12402 T^{5} + 96387 T^{6} - 22592 p T^{7} + 1715621 T^{8} - 3107483 T^{9} + 22553162 T^{10} - 20117519 T^{11} + 231881226 T^{12} + 67176915 T^{13} + 2074874657 T^{14} + 293127012 p T^{15} + 20170399972 T^{16} + 293127012 p^{2} T^{17} + 2074874657 p^{2} T^{18} + 67176915 p^{3} T^{19} + 231881226 p^{4} T^{20} - 20117519 p^{5} T^{21} + 22553162 p^{6} T^{22} - 3107483 p^{7} T^{23} + 1715621 p^{8} T^{24} - 22592 p^{10} T^{25} + 96387 p^{10} T^{26} - 12402 p^{11} T^{27} + 3750 p^{12} T^{28} - 368 p^{13} T^{29} + 90 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 6 T + 97 T^{2} - 359 T^{3} + 3862 T^{4} - 8739 T^{5} + 97576 T^{6} - 109826 T^{7} + 1948915 T^{8} - 237632 T^{9} + 34316745 T^{10} + 20907884 T^{11} + 569527538 T^{12} + 521147076 T^{13} + 8794275353 T^{14} + 8484394754 T^{15} + 121320902098 T^{16} + 8484394754 p T^{17} + 8794275353 p^{2} T^{18} + 521147076 p^{3} T^{19} + 569527538 p^{4} T^{20} + 20907884 p^{5} T^{21} + 34316745 p^{6} T^{22} - 237632 p^{7} T^{23} + 1948915 p^{8} T^{24} - 109826 p^{9} T^{25} + 97576 p^{10} T^{26} - 8739 p^{11} T^{27} + 3862 p^{12} T^{28} - 359 p^{13} T^{29} + 97 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 3 T + 83 T^{2} - 125 T^{3} + 2861 T^{4} - 992 T^{5} + 69519 T^{6} - 20198 T^{7} + 1771472 T^{8} - 1756390 T^{9} + 39865506 T^{10} - 38769572 T^{11} + 711027721 T^{12} - 699954287 T^{13} + 13151255436 T^{14} - 1154426801 p T^{15} + 245439220562 T^{16} - 1154426801 p^{2} T^{17} + 13151255436 p^{2} T^{18} - 699954287 p^{3} T^{19} + 711027721 p^{4} T^{20} - 38769572 p^{5} T^{21} + 39865506 p^{6} T^{22} - 1756390 p^{7} T^{23} + 1771472 p^{8} T^{24} - 20198 p^{9} T^{25} + 69519 p^{10} T^{26} - 992 p^{11} T^{27} + 2861 p^{12} T^{28} - 125 p^{13} T^{29} + 83 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 11 T + 210 T^{2} - 1815 T^{3} + 20282 T^{4} - 145097 T^{5} + 1217138 T^{6} - 7441281 T^{7} + 51415617 T^{8} - 274818036 T^{9} + 1645993344 T^{10} - 7853002292 T^{11} + 42366418590 T^{12} - 184882176290 T^{13} + 930230079980 T^{14} - 3820111749718 T^{15} + 18399434920492 T^{16} - 3820111749718 p T^{17} + 930230079980 p^{2} T^{18} - 184882176290 p^{3} T^{19} + 42366418590 p^{4} T^{20} - 7853002292 p^{5} T^{21} + 1645993344 p^{6} T^{22} - 274818036 p^{7} T^{23} + 51415617 p^{8} T^{24} - 7441281 p^{9} T^{25} + 1217138 p^{10} T^{26} - 145097 p^{11} T^{27} + 20282 p^{12} T^{28} - 1815 p^{13} T^{29} + 210 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 14 T + 294 T^{2} - 3265 T^{3} + 42055 T^{4} - 395598 T^{5} + 3968314 T^{6} - 32714604 T^{7} + 278210372 T^{8} - 2057052635 T^{9} + 15470768158 T^{10} - 104234601543 T^{11} + 710239861465 T^{12} - 4407340071450 T^{13} + 27603613889578 T^{14} - 158782070053245 T^{15} + 921279848105094 T^{16} - 158782070053245 p T^{17} + 27603613889578 p^{2} T^{18} - 4407340071450 p^{3} T^{19} + 710239861465 p^{4} T^{20} - 104234601543 p^{5} T^{21} + 15470768158 p^{6} T^{22} - 2057052635 p^{7} T^{23} + 278210372 p^{8} T^{24} - 32714604 p^{9} T^{25} + 3968314 p^{10} T^{26} - 395598 p^{11} T^{27} + 42055 p^{12} T^{28} - 3265 p^{13} T^{29} + 294 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 4 T + 348 T^{2} - 1381 T^{3} + 60800 T^{4} - 246520 T^{5} + 7069922 T^{6} - 29586173 T^{7} + 612259255 T^{8} - 71235411 p T^{9} + 41930613664 T^{10} - 183054493959 T^{11} + 2355141441220 T^{12} - 10182672001961 T^{13} + 110988909462834 T^{14} - 460428621018917 T^{15} + 4445235835686068 T^{16} - 460428621018917 p T^{17} + 110988909462834 p^{2} T^{18} - 10182672001961 p^{3} T^{19} + 2355141441220 p^{4} T^{20} - 183054493959 p^{5} T^{21} + 41930613664 p^{6} T^{22} - 71235411 p^{8} T^{23} + 612259255 p^{8} T^{24} - 29586173 p^{9} T^{25} + 7069922 p^{10} T^{26} - 246520 p^{11} T^{27} + 60800 p^{12} T^{28} - 1381 p^{13} T^{29} + 348 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 11 T + 351 T^{2} - 3509 T^{3} + 64654 T^{4} - 14049 p T^{5} + 7980108 T^{6} - 64037115 T^{7} + 735336275 T^{8} - 5364634568 T^{9} + 53746803598 T^{10} - 359761312208 T^{11} + 3241410212330 T^{12} - 20060735863402 T^{13} + 165714106021951 T^{14} - 953277968669194 T^{15} + 7299109474364504 T^{16} - 953277968669194 p T^{17} + 165714106021951 p^{2} T^{18} - 20060735863402 p^{3} T^{19} + 3241410212330 p^{4} T^{20} - 359761312208 p^{5} T^{21} + 53746803598 p^{6} T^{22} - 5364634568 p^{7} T^{23} + 735336275 p^{8} T^{24} - 64037115 p^{9} T^{25} + 7980108 p^{10} T^{26} - 14049 p^{12} T^{27} + 64654 p^{12} T^{28} - 3509 p^{13} T^{29} + 351 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 23 T + 621 T^{2} - 10629 T^{3} + 176736 T^{4} - 2424625 T^{5} + 31250850 T^{6} - 360299505 T^{7} + 3893683705 T^{8} - 38882456890 T^{9} + 365372984754 T^{10} - 3221700793654 T^{11} + 26840051705792 T^{12} - 211481434110410 T^{13} + 1578864928177479 T^{14} - 11191510143979808 T^{15} + 75271909946487532 T^{16} - 11191510143979808 p T^{17} + 1578864928177479 p^{2} T^{18} - 211481434110410 p^{3} T^{19} + 26840051705792 p^{4} T^{20} - 3221700793654 p^{5} T^{21} + 365372984754 p^{6} T^{22} - 38882456890 p^{7} T^{23} + 3893683705 p^{8} T^{24} - 360299505 p^{9} T^{25} + 31250850 p^{10} T^{26} - 2424625 p^{11} T^{27} + 176736 p^{12} T^{28} - 10629 p^{13} T^{29} + 621 p^{14} T^{30} - 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 2 T + 318 T^{2} - 4 T^{3} + 50079 T^{4} - 77447 T^{5} + 5344865 T^{6} - 14968679 T^{7} + 432946692 T^{8} - 1678285661 T^{9} + 28484569213 T^{10} - 136345336793 T^{11} + 1603599668977 T^{12} - 8855144414230 T^{13} + 81258028210116 T^{14} - 483256164976500 T^{15} + 3890208356154710 T^{16} - 483256164976500 p T^{17} + 81258028210116 p^{2} T^{18} - 8855144414230 p^{3} T^{19} + 1603599668977 p^{4} T^{20} - 136345336793 p^{5} T^{21} + 28484569213 p^{6} T^{22} - 1678285661 p^{7} T^{23} + 432946692 p^{8} T^{24} - 14968679 p^{9} T^{25} + 5344865 p^{10} T^{26} - 77447 p^{11} T^{27} + 50079 p^{12} T^{28} - 4 p^{13} T^{29} + 318 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 19 T + 431 T^{2} - 5819 T^{3} + 85523 T^{4} - 17934 p T^{5} + 11164835 T^{6} - 107392850 T^{7} + 1081832710 T^{8} - 9255764980 T^{9} + 83301576533 T^{10} - 649451637176 T^{11} + 5397180784669 T^{12} - 39327106176827 T^{13} + 311239365622937 T^{14} - 2179312691515715 T^{15} + 16832485566076786 T^{16} - 2179312691515715 p T^{17} + 311239365622937 p^{2} T^{18} - 39327106176827 p^{3} T^{19} + 5397180784669 p^{4} T^{20} - 649451637176 p^{5} T^{21} + 83301576533 p^{6} T^{22} - 9255764980 p^{7} T^{23} + 1081832710 p^{8} T^{24} - 107392850 p^{9} T^{25} + 11164835 p^{10} T^{26} - 17934 p^{12} T^{27} + 85523 p^{12} T^{28} - 5819 p^{13} T^{29} + 431 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 32 T + 822 T^{2} - 16290 T^{3} + 280616 T^{4} - 4266252 T^{5} + 59002420 T^{6} - 750388520 T^{7} + 8889961715 T^{8} - 98683415146 T^{9} + 1034078840112 T^{10} - 10260068417802 T^{11} + 96817508417900 T^{12} - 870380894105704 T^{13} + 7473128002490358 T^{14} - 61327399691512338 T^{15} + 481586259456684752 T^{16} - 61327399691512338 p T^{17} + 7473128002490358 p^{2} T^{18} - 870380894105704 p^{3} T^{19} + 96817508417900 p^{4} T^{20} - 10260068417802 p^{5} T^{21} + 1034078840112 p^{6} T^{22} - 98683415146 p^{7} T^{23} + 8889961715 p^{8} T^{24} - 750388520 p^{9} T^{25} + 59002420 p^{10} T^{26} - 4266252 p^{11} T^{27} + 280616 p^{12} T^{28} - 16290 p^{13} T^{29} + 822 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 19 T + 724 T^{2} - 12046 T^{3} + 257102 T^{4} - 3745752 T^{5} + 59115952 T^{6} - 758558610 T^{7} + 9828612801 T^{8} - 112001359983 T^{9} + 1252735431022 T^{10} - 12783502348115 T^{11} + 126737259090406 T^{12} - 1166087086221651 T^{13} + 10393498646127598 T^{14} - 86613947283856460 T^{15} + 698885105370952532 T^{16} - 86613947283856460 p T^{17} + 10393498646127598 p^{2} T^{18} - 1166087086221651 p^{3} T^{19} + 126737259090406 p^{4} T^{20} - 12783502348115 p^{5} T^{21} + 1252735431022 p^{6} T^{22} - 112001359983 p^{7} T^{23} + 9828612801 p^{8} T^{24} - 758558610 p^{9} T^{25} + 59115952 p^{10} T^{26} - 3745752 p^{11} T^{27} + 257102 p^{12} T^{28} - 12046 p^{13} T^{29} + 724 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 33 T + 949 T^{2} - 18749 T^{3} + 341207 T^{4} - 5197800 T^{5} + 75073803 T^{6} - 969967468 T^{7} + 12034150648 T^{8} - 137435325418 T^{9} + 1515871577664 T^{10} - 15620337250128 T^{11} + 156015993236885 T^{12} - 1468272486068645 T^{13} + 13422503694450576 T^{14} - 116108234925557999 T^{15} + 976415923955764262 T^{16} - 116108234925557999 p T^{17} + 13422503694450576 p^{2} T^{18} - 1468272486068645 p^{3} T^{19} + 156015993236885 p^{4} T^{20} - 15620337250128 p^{5} T^{21} + 1515871577664 p^{6} T^{22} - 137435325418 p^{7} T^{23} + 12034150648 p^{8} T^{24} - 969967468 p^{9} T^{25} + 75073803 p^{10} T^{26} - 5197800 p^{11} T^{27} + 341207 p^{12} T^{28} - 18749 p^{13} T^{29} + 949 p^{14} T^{30} - 33 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 5 T + 524 T^{2} + 2352 T^{3} + 149201 T^{4} + 595217 T^{5} + 29572318 T^{6} + 103995194 T^{7} + 4508719255 T^{8} + 13939173581 T^{9} + 557562404560 T^{10} + 1520509439145 T^{11} + 57725098788352 T^{12} + 140465013275417 T^{13} + 5102567896652094 T^{14} + 11293730464414365 T^{15} + 389271182680022622 T^{16} + 11293730464414365 p T^{17} + 5102567896652094 p^{2} T^{18} + 140465013275417 p^{3} T^{19} + 57725098788352 p^{4} T^{20} + 1520509439145 p^{5} T^{21} + 557562404560 p^{6} T^{22} + 13939173581 p^{7} T^{23} + 4508719255 p^{8} T^{24} + 103995194 p^{9} T^{25} + 29572318 p^{10} T^{26} + 595217 p^{11} T^{27} + 149201 p^{12} T^{28} + 2352 p^{13} T^{29} + 524 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 23 T + 1033 T^{2} - 19008 T^{3} + 492054 T^{4} - 7612833 T^{5} + 146285379 T^{6} - 1961092652 T^{7} + 30725787368 T^{8} - 363799575798 T^{9} + 4869148361641 T^{10} - 51538421046465 T^{11} + 604954307644026 T^{12} - 5765014485858262 T^{13} + 60276142525015787 T^{14} - 518759265856954575 T^{15} + 4874762472412736270 T^{16} - 518759265856954575 p T^{17} + 60276142525015787 p^{2} T^{18} - 5765014485858262 p^{3} T^{19} + 604954307644026 p^{4} T^{20} - 51538421046465 p^{5} T^{21} + 4869148361641 p^{6} T^{22} - 363799575798 p^{7} T^{23} + 30725787368 p^{8} T^{24} - 1961092652 p^{9} T^{25} + 146285379 p^{10} T^{26} - 7612833 p^{11} T^{27} + 492054 p^{12} T^{28} - 19008 p^{13} T^{29} + 1033 p^{14} T^{30} - 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 24 T + 976 T^{2} - 17809 T^{3} + 423439 T^{4} - 6352844 T^{5} + 113351534 T^{6} - 1460320242 T^{7} + 271957532 p T^{8} - 244361185721 T^{9} + 3116047394294 T^{10} - 31936971160069 T^{11} + 364590920008457 T^{12} - 3423996579040382 T^{13} + 35802802462447500 T^{14} - 311998492413383683 T^{15} + 3031192079903180294 T^{16} - 311998492413383683 p T^{17} + 35802802462447500 p^{2} T^{18} - 3423996579040382 p^{3} T^{19} + 364590920008457 p^{4} T^{20} - 31936971160069 p^{5} T^{21} + 3116047394294 p^{6} T^{22} - 244361185721 p^{7} T^{23} + 271957532 p^{9} T^{24} - 1460320242 p^{9} T^{25} + 113351534 p^{10} T^{26} - 6352844 p^{11} T^{27} + 423439 p^{12} T^{28} - 17809 p^{13} T^{29} + 976 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 7 T + 721 T^{2} - 5329 T^{3} + 268417 T^{4} - 1996258 T^{5} + 67821703 T^{6} - 490771140 T^{7} + 12924366426 T^{8} - 89194290838 T^{9} + 1962515432901 T^{10} - 12771659562060 T^{11} + 245541472047935 T^{12} - 1496562645439495 T^{13} + 25858985819788019 T^{14} - 146790708985858061 T^{15} + 27957566394452206 p T^{16} - 146790708985858061 p T^{17} + 25858985819788019 p^{2} T^{18} - 1496562645439495 p^{3} T^{19} + 245541472047935 p^{4} T^{20} - 12771659562060 p^{5} T^{21} + 1962515432901 p^{6} T^{22} - 89194290838 p^{7} T^{23} + 12924366426 p^{8} T^{24} - 490771140 p^{9} T^{25} + 67821703 p^{10} T^{26} - 1996258 p^{11} T^{27} + 268417 p^{12} T^{28} - 5329 p^{13} T^{29} + 721 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 2 T + 737 T^{2} + 1433 T^{3} + 278119 T^{4} + 460613 T^{5} + 71322669 T^{6} + 92702047 T^{7} + 13910198950 T^{8} + 13584922067 T^{9} + 2189781117450 T^{10} + 1600640366051 T^{11} + 287863420852619 T^{12} + 164145436153087 T^{13} + 32189395580626840 T^{14} + 15491547806714788 T^{15} + 3088744141027715150 T^{16} + 15491547806714788 p T^{17} + 32189395580626840 p^{2} T^{18} + 164145436153087 p^{3} T^{19} + 287863420852619 p^{4} T^{20} + 1600640366051 p^{5} T^{21} + 2189781117450 p^{6} T^{22} + 13584922067 p^{7} T^{23} + 13910198950 p^{8} T^{24} + 92702047 p^{9} T^{25} + 71322669 p^{10} T^{26} + 460613 p^{11} T^{27} + 278119 p^{12} T^{28} + 1433 p^{13} T^{29} + 737 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 33 T + 1684 T^{2} - 42506 T^{3} + 1258657 T^{4} - 25952915 T^{5} + 570633520 T^{6} - 9975451732 T^{7} + 178170946482 T^{8} - 2701117024970 T^{9} + 40944616452912 T^{10} - 545983773475415 T^{11} + 7196188837938271 T^{12} - 85124759399088460 T^{13} + 989278876034505676 T^{14} - 10425323056413780161 T^{15} + \)\(10\!\cdots\!10\)\( T^{16} - 10425323056413780161 p T^{17} + 989278876034505676 p^{2} T^{18} - 85124759399088460 p^{3} T^{19} + 7196188837938271 p^{4} T^{20} - 545983773475415 p^{5} T^{21} + 40944616452912 p^{6} T^{22} - 2701117024970 p^{7} T^{23} + 178170946482 p^{8} T^{24} - 9975451732 p^{9} T^{25} + 570633520 p^{10} T^{26} - 25952915 p^{11} T^{27} + 1258657 p^{12} T^{28} - 42506 p^{13} T^{29} + 1684 p^{14} T^{30} - 33 p^{15} T^{31} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.85434725800220879732006596321, −1.80459474103182268544735751075, −1.77301526862498186376390551942, −1.76372631882040785730639114767, −1.70764937931677192887086454240, −1.69446228610328820895965450073, −1.69204906060381066099086875400, −1.68835552744437842029053233958, −1.58515281328143001412263580175, −1.20269040816193260036434779832, −1.11773143102213837758867721092, −1.08445093557216603842412879363, −1.06222629920341750577963249055, −1.03110399521854255839071144618, −1.02082646472062891868015763374, −0.976245152710285356952472904668, −0.840156139820717569782745286461, −0.821400598416738282498528385174, −0.77111572114126963912441107576, −0.74735741452627708571588207390, −0.62398524502311777304020449595, −0.60932525063701561510798155696, −0.39071070557498265506708825896, −0.37519141919393307256627573764, −0.36840260269670983551651814595, 0.36840260269670983551651814595, 0.37519141919393307256627573764, 0.39071070557498265506708825896, 0.60932525063701561510798155696, 0.62398524502311777304020449595, 0.74735741452627708571588207390, 0.77111572114126963912441107576, 0.821400598416738282498528385174, 0.840156139820717569782745286461, 0.976245152710285356952472904668, 1.02082646472062891868015763374, 1.03110399521854255839071144618, 1.06222629920341750577963249055, 1.08445093557216603842412879363, 1.11773143102213837758867721092, 1.20269040816193260036434779832, 1.58515281328143001412263580175, 1.68835552744437842029053233958, 1.69204906060381066099086875400, 1.69446228610328820895965450073, 1.70764937931677192887086454240, 1.76372631882040785730639114767, 1.77301526862498186376390551942, 1.80459474103182268544735751075, 1.85434725800220879732006596321
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.