Dirichlet series
| L(s) = 1 | − 15·2-s + 120·4-s − 5-s − 18·7-s − 680·8-s − 14·9-s + 15·10-s − 4·11-s + 270·14-s + 3.06e3·16-s + 2·17-s + 210·18-s + 15·19-s − 120·20-s + 60·22-s + 17·23-s − 33·25-s + 4·27-s − 2.16e3·28-s − 20·29-s − 30·31-s − 1.16e4·32-s − 30·34-s + 18·35-s − 1.68e3·36-s − 35·37-s − 225·38-s + ⋯ |
| L(s) = 1 | − 10.6·2-s + 60·4-s − 0.447·5-s − 6.80·7-s − 240.·8-s − 4.66·9-s + 4.74·10-s − 1.20·11-s + 72.1·14-s + 765·16-s + 0.485·17-s + 49.4·18-s + 3.44·19-s − 26.8·20-s + 12.7·22-s + 3.54·23-s − 6.59·25-s + 0.769·27-s − 408.·28-s − 3.71·29-s − 5.38·31-s − 2.05e3·32-s − 5.14·34-s + 3.04·35-s − 280·36-s − 5.75·37-s − 36.4·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 13^{30} \cdot 19^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 13^{30} \cdot 19^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{15} \cdot 13^{30} \cdot 19^{15}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.45872\times 10^{25}\) |
| Root analytic conductor: | \(7.16100\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(15\) |
| Selberg data: | \((30,\ 2^{15} \cdot 13^{30} \cdot 19^{15} ,\ ( \ : [1/2]^{15} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{15} \) |
| 13 | \( 1 \) | |
| 19 | \( ( 1 - T )^{15} \) | |
| good | 3 | \( 1 + 14 T^{2} - 4 T^{3} + 109 T^{4} - 59 T^{5} + 208 p T^{6} - 154 p T^{7} + 2993 T^{8} - 2594 T^{9} + 1391 p^{2} T^{10} - 11677 T^{11} + 15413 p T^{12} - 44425 T^{13} + 50903 p T^{14} - 144046 T^{15} + 50903 p^{2} T^{16} - 44425 p^{2} T^{17} + 15413 p^{4} T^{18} - 11677 p^{4} T^{19} + 1391 p^{7} T^{20} - 2594 p^{6} T^{21} + 2993 p^{7} T^{22} - 154 p^{9} T^{23} + 208 p^{10} T^{24} - 59 p^{10} T^{25} + 109 p^{11} T^{26} - 4 p^{12} T^{27} + 14 p^{13} T^{28} + p^{15} T^{30} \) |
| 5 | \( 1 + T + 34 T^{2} + 32 T^{3} + 626 T^{4} + 22 p^{2} T^{5} + 8063 T^{6} + 1313 p T^{7} + 80536 T^{8} + 12144 p T^{9} + 131273 p T^{10} + 18337 p^{2} T^{11} + 897796 p T^{12} + 581773 p T^{13} + 26159177 T^{14} + 15721783 T^{15} + 26159177 p T^{16} + 581773 p^{3} T^{17} + 897796 p^{4} T^{18} + 18337 p^{6} T^{19} + 131273 p^{6} T^{20} + 12144 p^{7} T^{21} + 80536 p^{7} T^{22} + 1313 p^{9} T^{23} + 8063 p^{9} T^{24} + 22 p^{12} T^{25} + 626 p^{11} T^{26} + 32 p^{12} T^{27} + 34 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 + 18 T + 200 T^{2} + 1639 T^{3} + 10925 T^{4} + 61993 T^{5} + 309200 T^{6} + 1384776 T^{7} + 5661343 T^{8} + 21394830 T^{9} + 75484708 T^{10} + 250529448 T^{11} + 786663536 T^{12} + 2346178489 T^{13} + 951973283 p T^{14} + 18049598790 T^{15} + 951973283 p^{2} T^{16} + 2346178489 p^{2} T^{17} + 786663536 p^{3} T^{18} + 250529448 p^{4} T^{19} + 75484708 p^{5} T^{20} + 21394830 p^{6} T^{21} + 5661343 p^{7} T^{22} + 1384776 p^{8} T^{23} + 309200 p^{9} T^{24} + 61993 p^{10} T^{25} + 10925 p^{11} T^{26} + 1639 p^{12} T^{27} + 200 p^{13} T^{28} + 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 11 | \( 1 + 4 T + 71 T^{2} + 21 p T^{3} + 2474 T^{4} + 6430 T^{5} + 57326 T^{6} + 122706 T^{7} + 94812 p T^{8} + 1949592 T^{9} + 16385389 T^{10} + 28171631 T^{11} + 228437691 T^{12} + 368573804 T^{13} + 256937484 p T^{14} + 4296596972 T^{15} + 256937484 p^{2} T^{16} + 368573804 p^{2} T^{17} + 228437691 p^{3} T^{18} + 28171631 p^{4} T^{19} + 16385389 p^{5} T^{20} + 1949592 p^{6} T^{21} + 94812 p^{8} T^{22} + 122706 p^{8} T^{23} + 57326 p^{9} T^{24} + 6430 p^{10} T^{25} + 2474 p^{11} T^{26} + 21 p^{13} T^{27} + 71 p^{13} T^{28} + 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 - 2 T + 112 T^{2} - 150 T^{3} + 356 p T^{4} - 2981 T^{5} + 213941 T^{6} + 95347 T^{7} + 5825443 T^{8} + 7286442 T^{9} + 138226255 T^{10} + 225112534 T^{11} + 3023658614 T^{12} + 4770636769 T^{13} + 59516931621 T^{14} + 84887921937 T^{15} + 59516931621 p T^{16} + 4770636769 p^{2} T^{17} + 3023658614 p^{3} T^{18} + 225112534 p^{4} T^{19} + 138226255 p^{5} T^{20} + 7286442 p^{6} T^{21} + 5825443 p^{7} T^{22} + 95347 p^{8} T^{23} + 213941 p^{9} T^{24} - 2981 p^{10} T^{25} + 356 p^{12} T^{26} - 150 p^{12} T^{27} + 112 p^{13} T^{28} - 2 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 - 17 T + 274 T^{2} - 2994 T^{3} + 30660 T^{4} - 263089 T^{5} + 2138738 T^{6} - 15613823 T^{7} + 108942740 T^{8} - 703355132 T^{9} + 4364985334 T^{10} - 25428468436 T^{11} + 142916170295 T^{12} - 759471649018 T^{13} + 3902857199226 T^{14} - 19025437256614 T^{15} + 3902857199226 p T^{16} - 759471649018 p^{2} T^{17} + 142916170295 p^{3} T^{18} - 25428468436 p^{4} T^{19} + 4364985334 p^{5} T^{20} - 703355132 p^{6} T^{21} + 108942740 p^{7} T^{22} - 15613823 p^{8} T^{23} + 2138738 p^{9} T^{24} - 263089 p^{10} T^{25} + 30660 p^{11} T^{26} - 2994 p^{12} T^{27} + 274 p^{13} T^{28} - 17 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 + 20 T + 362 T^{2} + 4436 T^{3} + 50557 T^{4} + 479821 T^{5} + 4312624 T^{6} + 34597407 T^{7} + 266167139 T^{8} + 1890611122 T^{9} + 12989111224 T^{10} + 83811654918 T^{11} + 524950724341 T^{12} + 3112005474141 T^{13} + 17921794563648 T^{14} + 97932314027054 T^{15} + 17921794563648 p T^{16} + 3112005474141 p^{2} T^{17} + 524950724341 p^{3} T^{18} + 83811654918 p^{4} T^{19} + 12989111224 p^{5} T^{20} + 1890611122 p^{6} T^{21} + 266167139 p^{7} T^{22} + 34597407 p^{8} T^{23} + 4312624 p^{9} T^{24} + 479821 p^{10} T^{25} + 50557 p^{11} T^{26} + 4436 p^{12} T^{27} + 362 p^{13} T^{28} + 20 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + 30 T + 638 T^{2} + 9874 T^{3} + 131100 T^{4} + 1494467 T^{5} + 15471754 T^{6} + 144883403 T^{7} + 1261353090 T^{8} + 10164112970 T^{9} + 77085437089 T^{10} + 547738044779 T^{11} + 3687931223002 T^{12} + 23411645619631 T^{13} + 141308233325172 T^{14} + 806426947428548 T^{15} + 141308233325172 p T^{16} + 23411645619631 p^{2} T^{17} + 3687931223002 p^{3} T^{18} + 547738044779 p^{4} T^{19} + 77085437089 p^{5} T^{20} + 10164112970 p^{6} T^{21} + 1261353090 p^{7} T^{22} + 144883403 p^{8} T^{23} + 15471754 p^{9} T^{24} + 1494467 p^{10} T^{25} + 131100 p^{11} T^{26} + 9874 p^{12} T^{27} + 638 p^{13} T^{28} + 30 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 + 35 T + 900 T^{2} + 16527 T^{3} + 255846 T^{4} + 3326587 T^{5} + 38445574 T^{6} + 395119568 T^{7} + 3718745153 T^{8} + 32096223136 T^{9} + 258936956155 T^{10} + 1954785075383 T^{11} + 13999026028871 T^{12} + 95069086892490 T^{13} + 618109456563016 T^{14} + 3836610105532724 T^{15} + 618109456563016 p T^{16} + 95069086892490 p^{2} T^{17} + 13999026028871 p^{3} T^{18} + 1954785075383 p^{4} T^{19} + 258936956155 p^{5} T^{20} + 32096223136 p^{6} T^{21} + 3718745153 p^{7} T^{22} + 395119568 p^{8} T^{23} + 38445574 p^{9} T^{24} + 3326587 p^{10} T^{25} + 255846 p^{11} T^{26} + 16527 p^{12} T^{27} + 900 p^{13} T^{28} + 35 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 15 T + 577 T^{2} + 7168 T^{3} + 3669 p T^{4} + 1593552 T^{5} + 23838307 T^{6} + 219861657 T^{7} + 2601747543 T^{8} + 21226692667 T^{9} + 210116312557 T^{10} + 1535315042450 T^{11} + 13172160762789 T^{12} + 86932604070454 T^{13} + 16148816552093 p T^{14} + 3957112035603594 T^{15} + 16148816552093 p^{2} T^{16} + 86932604070454 p^{2} T^{17} + 13172160762789 p^{3} T^{18} + 1535315042450 p^{4} T^{19} + 210116312557 p^{5} T^{20} + 21226692667 p^{6} T^{21} + 2601747543 p^{7} T^{22} + 219861657 p^{8} T^{23} + 23838307 p^{9} T^{24} + 1593552 p^{10} T^{25} + 3669 p^{12} T^{26} + 7168 p^{12} T^{27} + 577 p^{13} T^{28} + 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 - T + 353 T^{2} - 1101 T^{3} + 61403 T^{4} - 311520 T^{5} + 7196812 T^{6} - 47852116 T^{7} + 646579399 T^{8} - 4942096773 T^{9} + 47241578767 T^{10} - 378608199523 T^{11} + 2875038802820 T^{12} - 22643517120176 T^{13} + 146908922061155 T^{14} - 1084134867911380 T^{15} + 146908922061155 p T^{16} - 22643517120176 p^{2} T^{17} + 2875038802820 p^{3} T^{18} - 378608199523 p^{4} T^{19} + 47241578767 p^{5} T^{20} - 4942096773 p^{6} T^{21} + 646579399 p^{7} T^{22} - 47852116 p^{8} T^{23} + 7196812 p^{9} T^{24} - 311520 p^{10} T^{25} + 61403 p^{11} T^{26} - 1101 p^{12} T^{27} + 353 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + 147 T^{2} + 549 T^{3} + 15243 T^{4} + 55718 T^{5} + 1381452 T^{6} + 4477976 T^{7} + 90223209 T^{8} + 298982547 T^{9} + 5293164754 T^{10} + 13762253173 T^{11} + 277665096294 T^{12} + 13291442297 p T^{13} + 13218605024826 T^{14} + 28845884624380 T^{15} + 13218605024826 p T^{16} + 13291442297 p^{3} T^{17} + 277665096294 p^{3} T^{18} + 13762253173 p^{4} T^{19} + 5293164754 p^{5} T^{20} + 298982547 p^{6} T^{21} + 90223209 p^{7} T^{22} + 4477976 p^{8} T^{23} + 1381452 p^{9} T^{24} + 55718 p^{10} T^{25} + 15243 p^{11} T^{26} + 549 p^{12} T^{27} + 147 p^{13} T^{28} + p^{15} T^{30} \) | |
| 53 | \( 1 + T + 226 T^{2} - 345 T^{3} + 29968 T^{4} - 87436 T^{5} + 3167103 T^{6} - 10622721 T^{7} + 280942513 T^{8} - 976602577 T^{9} + 21098939746 T^{10} - 75037901621 T^{11} + 1391451096024 T^{12} - 4815007783748 T^{13} + 82579871838123 T^{14} - 269067554954274 T^{15} + 82579871838123 p T^{16} - 4815007783748 p^{2} T^{17} + 1391451096024 p^{3} T^{18} - 75037901621 p^{4} T^{19} + 21098939746 p^{5} T^{20} - 976602577 p^{6} T^{21} + 280942513 p^{7} T^{22} - 10622721 p^{8} T^{23} + 3167103 p^{9} T^{24} - 87436 p^{10} T^{25} + 29968 p^{11} T^{26} - 345 p^{12} T^{27} + 226 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 - 7 T + 447 T^{2} - 2296 T^{3} + 98204 T^{4} - 378358 T^{5} + 14581505 T^{6} - 42491999 T^{7} + 1664083746 T^{8} - 3723032214 T^{9} + 155787805558 T^{10} - 277910955570 T^{11} + 12367389898346 T^{12} - 18620998949793 T^{13} + 844342779127127 T^{14} - 1144115458090542 T^{15} + 844342779127127 p T^{16} - 18620998949793 p^{2} T^{17} + 12367389898346 p^{3} T^{18} - 277910955570 p^{4} T^{19} + 155787805558 p^{5} T^{20} - 3723032214 p^{6} T^{21} + 1664083746 p^{7} T^{22} - 42491999 p^{8} T^{23} + 14581505 p^{9} T^{24} - 378358 p^{10} T^{25} + 98204 p^{11} T^{26} - 2296 p^{12} T^{27} + 447 p^{13} T^{28} - 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 2 T + 259 T^{2} + 403 T^{3} + 43186 T^{4} + 71816 T^{5} + 5540352 T^{6} + 9986317 T^{7} + 582663335 T^{8} + 1055619776 T^{9} + 52244062858 T^{10} + 95749589845 T^{11} + 4101918860010 T^{12} + 7273395441016 T^{13} + 282504920327780 T^{14} + 474672726877519 T^{15} + 282504920327780 p T^{16} + 7273395441016 p^{2} T^{17} + 4101918860010 p^{3} T^{18} + 95749589845 p^{4} T^{19} + 52244062858 p^{5} T^{20} + 1055619776 p^{6} T^{21} + 582663335 p^{7} T^{22} + 9986317 p^{8} T^{23} + 5540352 p^{9} T^{24} + 71816 p^{10} T^{25} + 43186 p^{11} T^{26} + 403 p^{12} T^{27} + 259 p^{13} T^{28} + 2 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 + 34 T + 1159 T^{2} + 24690 T^{3} + 504872 T^{4} + 8120737 T^{5} + 125782298 T^{6} + 1666068181 T^{7} + 21406304121 T^{8} + 244981781648 T^{9} + 2732749807773 T^{10} + 27744604436256 T^{11} + 275221471009439 T^{12} + 2513790032921631 T^{13} + 22464669105445019 T^{14} + 185846766986356838 T^{15} + 22464669105445019 p T^{16} + 2513790032921631 p^{2} T^{17} + 275221471009439 p^{3} T^{18} + 27744604436256 p^{4} T^{19} + 2732749807773 p^{5} T^{20} + 244981781648 p^{6} T^{21} + 21406304121 p^{7} T^{22} + 1666068181 p^{8} T^{23} + 125782298 p^{9} T^{24} + 8120737 p^{10} T^{25} + 504872 p^{11} T^{26} + 24690 p^{12} T^{27} + 1159 p^{13} T^{28} + 34 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 + 4 T + 659 T^{2} + 33 p T^{3} + 207782 T^{4} + 638919 T^{5} + 42022433 T^{6} + 107613694 T^{7} + 6182856201 T^{8} + 12597402568 T^{9} + 713364267231 T^{10} + 1112325063444 T^{11} + 67963417859747 T^{12} + 81804708293703 T^{13} + 5538244883996828 T^{14} + 5728011391951090 T^{15} + 5538244883996828 p T^{16} + 81804708293703 p^{2} T^{17} + 67963417859747 p^{3} T^{18} + 1112325063444 p^{4} T^{19} + 713364267231 p^{5} T^{20} + 12597402568 p^{6} T^{21} + 6182856201 p^{7} T^{22} + 107613694 p^{8} T^{23} + 42022433 p^{9} T^{24} + 638919 p^{10} T^{25} + 207782 p^{11} T^{26} + 33 p^{13} T^{27} + 659 p^{13} T^{28} + 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 12 T + 703 T^{2} + 6204 T^{3} + 224854 T^{4} + 1477629 T^{5} + 45390832 T^{6} + 220105693 T^{7} + 6718357632 T^{8} + 23755749774 T^{9} + 795157240234 T^{10} + 2076224736096 T^{11} + 78946428698936 T^{12} + 161800938066413 T^{13} + 6706158571112564 T^{14} + 11989923870610759 T^{15} + 6706158571112564 p T^{16} + 161800938066413 p^{2} T^{17} + 78946428698936 p^{3} T^{18} + 2076224736096 p^{4} T^{19} + 795157240234 p^{5} T^{20} + 23755749774 p^{6} T^{21} + 6718357632 p^{7} T^{22} + 220105693 p^{8} T^{23} + 45390832 p^{9} T^{24} + 1477629 p^{10} T^{25} + 224854 p^{11} T^{26} + 6204 p^{12} T^{27} + 703 p^{13} T^{28} + 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 - 23 T + 793 T^{2} - 14494 T^{3} + 294893 T^{4} - 4443418 T^{5} + 69031289 T^{6} - 887453659 T^{7} + 11548112869 T^{8} - 130658943341 T^{9} + 1492562807995 T^{10} - 15254592771293 T^{11} + 157856856544234 T^{12} - 1487207122851956 T^{13} + 14257058949136364 T^{14} - 125460138273608216 T^{15} + 14257058949136364 p T^{16} - 1487207122851956 p^{2} T^{17} + 157856856544234 p^{3} T^{18} - 15254592771293 p^{4} T^{19} + 1492562807995 p^{5} T^{20} - 130658943341 p^{6} T^{21} + 11548112869 p^{7} T^{22} - 887453659 p^{8} T^{23} + 69031289 p^{9} T^{24} - 4443418 p^{10} T^{25} + 294893 p^{11} T^{26} - 14494 p^{12} T^{27} + 793 p^{13} T^{28} - 23 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 - 3 T + 614 T^{2} - 1991 T^{3} + 202245 T^{4} - 678149 T^{5} + 46274930 T^{6} - 155211342 T^{7} + 8111194111 T^{8} - 26616937358 T^{9} + 1144625563887 T^{10} - 3607308964991 T^{11} + 133665245077085 T^{12} - 397842325778158 T^{13} + 13114470470596099 T^{14} - 36249858147756680 T^{15} + 13114470470596099 p T^{16} - 397842325778158 p^{2} T^{17} + 133665245077085 p^{3} T^{18} - 3607308964991 p^{4} T^{19} + 1144625563887 p^{5} T^{20} - 26616937358 p^{6} T^{21} + 8111194111 p^{7} T^{22} - 155211342 p^{8} T^{23} + 46274930 p^{9} T^{24} - 678149 p^{10} T^{25} + 202245 p^{11} T^{26} - 1991 p^{12} T^{27} + 614 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 17 T + 679 T^{2} + 9599 T^{3} + 219335 T^{4} + 2729749 T^{5} + 47674356 T^{6} + 542231109 T^{7} + 8092228469 T^{8} + 85527594830 T^{9} + 1139320889102 T^{10} + 11222309638071 T^{11} + 135880232613424 T^{12} + 1247982004810988 T^{13} + 13921293878358018 T^{14} + 119355252984263978 T^{15} + 13921293878358018 p T^{16} + 1247982004810988 p^{2} T^{17} + 135880232613424 p^{3} T^{18} + 11222309638071 p^{4} T^{19} + 1139320889102 p^{5} T^{20} + 85527594830 p^{6} T^{21} + 8092228469 p^{7} T^{22} + 542231109 p^{8} T^{23} + 47674356 p^{9} T^{24} + 2729749 p^{10} T^{25} + 219335 p^{11} T^{26} + 9599 p^{12} T^{27} + 679 p^{13} T^{28} + 17 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 + 18 T + 737 T^{2} + 9527 T^{3} + 235387 T^{4} + 2325107 T^{5} + 46199820 T^{6} + 350376619 T^{7} + 6508656415 T^{8} + 36133905976 T^{9} + 730251623140 T^{10} + 2686190101263 T^{11} + 72231639365380 T^{12} + 159682855723555 T^{13} + 6905082713915792 T^{14} + 11321373015595230 T^{15} + 6905082713915792 p T^{16} + 159682855723555 p^{2} T^{17} + 72231639365380 p^{3} T^{18} + 2686190101263 p^{4} T^{19} + 730251623140 p^{5} T^{20} + 36133905976 p^{6} T^{21} + 6508656415 p^{7} T^{22} + 350376619 p^{8} T^{23} + 46199820 p^{9} T^{24} + 2325107 p^{10} T^{25} + 235387 p^{11} T^{26} + 9527 p^{12} T^{27} + 737 p^{13} T^{28} + 18 p^{14} T^{29} + p^{15} T^{30} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.27792676281433076856342498855, −2.22332542016260276304098023553, −2.21686004603988544603778759753, −2.12596126310693062207420882127, −2.10664587686515664380391057911, −2.01934240350280220250588831749, −1.97611466342774675151386757230, −1.88608530095745235234511186117, −1.86907494058827087567662594906, −1.84943314920453221174081478996, −1.84006065521314695648842236658, −1.61595241049805349074835655462, −1.58326577410376052645930227966, −1.49537496784302293871761799877, −1.35807318776102389149241092680, −1.32241036076569215903650005963, −1.26827398441715838280465959460, −1.14529209964374965378553416421, −1.11956018142269595391608769339, −1.03831140411724824594538285183, −1.01565741679024371558460375066, −0.985715178484143966737154026249, −0.842140471235616929538368583969, −0.832004161602891141080813851270, −0.74655880218868072522301015589, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.74655880218868072522301015589, 0.832004161602891141080813851270, 0.842140471235616929538368583969, 0.985715178484143966737154026249, 1.01565741679024371558460375066, 1.03831140411724824594538285183, 1.11956018142269595391608769339, 1.14529209964374965378553416421, 1.26827398441715838280465959460, 1.32241036076569215903650005963, 1.35807318776102389149241092680, 1.49537496784302293871761799877, 1.58326577410376052645930227966, 1.61595241049805349074835655462, 1.84006065521314695648842236658, 1.84943314920453221174081478996, 1.86907494058827087567662594906, 1.88608530095745235234511186117, 1.97611466342774675151386757230, 2.01934240350280220250588831749, 2.10664587686515664380391057911, 2.12596126310693062207420882127, 2.21686004603988544603778759753, 2.22332542016260276304098023553, 2.27792676281433076856342498855
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.