Dirichlet series
| L(s) = 1 | + 9·3-s − 9·4-s + 12·7-s + 2·8-s + 24·9-s + 6·11-s − 81·12-s + 36·16-s + 6·19-s + 108·21-s + 36·23-s + 18·24-s − 24·27-s − 108·28-s − 18·29-s − 18·32-s + 54·33-s − 216·36-s + 12·37-s − 18·41-s − 3·43-s − 54·44-s − 3·47-s + 324·48-s + 27·49-s + 24·56-s + 54·57-s + ⋯ |
| L(s) = 1 | + 5.19·3-s − 9/2·4-s + 4.53·7-s + 0.707·8-s + 8·9-s + 1.80·11-s − 23.3·12-s + 9·16-s + 1.37·19-s + 23.5·21-s + 7.50·23-s + 3.67·24-s − 4.61·27-s − 20.4·28-s − 3.34·29-s − 3.18·32-s + 9.40·33-s − 36·36-s + 1.97·37-s − 2.81·41-s − 0.457·43-s − 8.14·44-s − 0.437·47-s + 46.7·48-s + 27/7·49-s + 3.20·56-s + 7.15·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{30} \cdot 17^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{30} \cdot 17^{30}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(5^{30} \cdot 17^{30}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.61050\times 10^{26}\) |
| Root analytic conductor: | \(7.59551\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((30,\ 5^{30} \cdot 17^{30} ,\ ( \ : [1/2]^{15} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(241.1174416\) |
| \(L(\frac12)\) | \(\approx\) | \(241.1174416\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + 9 T^{2} - p T^{3} + 45 T^{4} - 9 p T^{5} + 43 p^{2} T^{6} - 81 T^{7} + 279 p T^{8} - 267 T^{9} + 195 p^{3} T^{10} - 747 T^{11} + 3835 T^{12} - 909 p T^{13} + 8481 T^{14} - 3869 T^{15} + 8481 p T^{16} - 909 p^{3} T^{17} + 3835 p^{3} T^{18} - 747 p^{4} T^{19} + 195 p^{8} T^{20} - 267 p^{6} T^{21} + 279 p^{8} T^{22} - 81 p^{8} T^{23} + 43 p^{11} T^{24} - 9 p^{11} T^{25} + 45 p^{11} T^{26} - p^{13} T^{27} + 9 p^{13} T^{28} + p^{15} T^{30} \) |
| 3 | \( 1 - p^{2} T + 19 p T^{2} - 91 p T^{3} + 367 p T^{4} - 1283 p T^{5} + 4012 p T^{6} - 3799 p^{2} T^{7} + 29839 p T^{8} - 218035 T^{9} + 166037 p T^{10} - 357896 p T^{11} + 2194226 T^{12} - 17558 p^{5} T^{13} + 879233 p^{2} T^{14} - 14020073 T^{15} + 879233 p^{3} T^{16} - 17558 p^{7} T^{17} + 2194226 p^{3} T^{18} - 357896 p^{5} T^{19} + 166037 p^{6} T^{20} - 218035 p^{6} T^{21} + 29839 p^{8} T^{22} - 3799 p^{10} T^{23} + 4012 p^{10} T^{24} - 1283 p^{11} T^{25} + 367 p^{12} T^{26} - 91 p^{13} T^{27} + 19 p^{14} T^{28} - p^{16} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 - 12 T + 117 T^{2} - 113 p T^{3} + 4698 T^{4} - 474 p^{2} T^{5} + 104992 T^{6} - 422754 T^{7} + 228525 p T^{8} - 5599859 T^{9} + 2681895 p T^{10} - 59503659 T^{11} + 182193565 T^{12} - 531877326 T^{13} + 1500614802 T^{14} - 4040728669 T^{15} + 1500614802 p T^{16} - 531877326 p^{2} T^{17} + 182193565 p^{3} T^{18} - 59503659 p^{4} T^{19} + 2681895 p^{6} T^{20} - 5599859 p^{6} T^{21} + 228525 p^{8} T^{22} - 422754 p^{8} T^{23} + 104992 p^{9} T^{24} - 474 p^{12} T^{25} + 4698 p^{11} T^{26} - 113 p^{13} T^{27} + 117 p^{13} T^{28} - 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 11 | \( 1 - 6 T + 117 T^{2} - 57 p T^{3} + 6507 T^{4} - 31560 T^{5} + 230570 T^{6} - 1021344 T^{7} + 5877747 T^{8} - 23942081 T^{9} + 10489278 p T^{10} - 434304060 T^{11} + 1822602619 T^{12} - 6349650663 T^{13} + 2167823442 p T^{14} - 76595406407 T^{15} + 2167823442 p^{2} T^{16} - 6349650663 p^{2} T^{17} + 1822602619 p^{3} T^{18} - 434304060 p^{4} T^{19} + 10489278 p^{6} T^{20} - 23942081 p^{6} T^{21} + 5877747 p^{7} T^{22} - 1021344 p^{8} T^{23} + 230570 p^{9} T^{24} - 31560 p^{10} T^{25} + 6507 p^{11} T^{26} - 57 p^{13} T^{27} + 117 p^{13} T^{28} - 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 + 69 T^{2} - 9 T^{3} + 2490 T^{4} - 900 T^{5} + 66348 T^{6} - 38901 T^{7} + 1455057 T^{8} - 1069283 T^{9} + 27203880 T^{10} - 22183884 T^{11} + 446426155 T^{12} - 372862257 T^{13} + 502059342 p T^{14} - 5253958989 T^{15} + 502059342 p^{2} T^{16} - 372862257 p^{2} T^{17} + 446426155 p^{3} T^{18} - 22183884 p^{4} T^{19} + 27203880 p^{5} T^{20} - 1069283 p^{6} T^{21} + 1455057 p^{7} T^{22} - 38901 p^{8} T^{23} + 66348 p^{9} T^{24} - 900 p^{10} T^{25} + 2490 p^{11} T^{26} - 9 p^{12} T^{27} + 69 p^{13} T^{28} + p^{15} T^{30} \) | |
| 19 | \( 1 - 6 T + 90 T^{2} - 574 T^{3} + 5133 T^{4} - 29607 T^{5} + 207133 T^{6} - 1099329 T^{7} + 6592116 T^{8} - 32102291 T^{9} + 172999752 T^{10} - 792321015 T^{11} + 3946731101 T^{12} - 17144140986 T^{13} + 81119671902 T^{14} - 339550234751 T^{15} + 81119671902 p T^{16} - 17144140986 p^{2} T^{17} + 3946731101 p^{3} T^{18} - 792321015 p^{4} T^{19} + 172999752 p^{5} T^{20} - 32102291 p^{6} T^{21} + 6592116 p^{7} T^{22} - 1099329 p^{8} T^{23} + 207133 p^{9} T^{24} - 29607 p^{10} T^{25} + 5133 p^{11} T^{26} - 574 p^{12} T^{27} + 90 p^{13} T^{28} - 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 - 36 T + 780 T^{2} - 12418 T^{3} + 160806 T^{4} - 1775169 T^{5} + 17252865 T^{6} - 150555408 T^{7} + 52051146 p T^{8} - 8761408073 T^{9} + 59474810166 T^{10} - 376525834737 T^{11} + 2232628241152 T^{12} - 12434611405029 T^{13} + 65185821349515 T^{14} - 321984517750125 T^{15} + 65185821349515 p T^{16} - 12434611405029 p^{2} T^{17} + 2232628241152 p^{3} T^{18} - 376525834737 p^{4} T^{19} + 59474810166 p^{5} T^{20} - 8761408073 p^{6} T^{21} + 52051146 p^{8} T^{22} - 150555408 p^{8} T^{23} + 17252865 p^{9} T^{24} - 1775169 p^{10} T^{25} + 160806 p^{11} T^{26} - 12418 p^{12} T^{27} + 780 p^{13} T^{28} - 36 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 + 18 T + 324 T^{2} + 3929 T^{3} + 45855 T^{4} + 439560 T^{5} + 4059906 T^{6} + 33041538 T^{7} + 260440050 T^{8} + 1868618587 T^{9} + 13050645729 T^{10} + 84454945560 T^{11} + 533883373741 T^{12} + 3157358006988 T^{13} + 18281753765688 T^{14} + 99462401538111 T^{15} + 18281753765688 p T^{16} + 3157358006988 p^{2} T^{17} + 533883373741 p^{3} T^{18} + 84454945560 p^{4} T^{19} + 13050645729 p^{5} T^{20} + 1868618587 p^{6} T^{21} + 260440050 p^{7} T^{22} + 33041538 p^{8} T^{23} + 4059906 p^{9} T^{24} + 439560 p^{10} T^{25} + 45855 p^{11} T^{26} + 3929 p^{12} T^{27} + 324 p^{13} T^{28} + 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + 255 T^{2} + 81 T^{3} + 31197 T^{4} + 26862 T^{5} + 2420465 T^{6} + 4074891 T^{7} + 133123248 T^{8} + 12275705 p T^{9} + 5556065769 T^{10} + 24580898337 T^{11} + 189000067330 T^{12} + 1162421834223 T^{13} + 5818260146034 T^{14} + 41367211082971 T^{15} + 5818260146034 p T^{16} + 1162421834223 p^{2} T^{17} + 189000067330 p^{3} T^{18} + 24580898337 p^{4} T^{19} + 5556065769 p^{5} T^{20} + 12275705 p^{7} T^{21} + 133123248 p^{7} T^{22} + 4074891 p^{8} T^{23} + 2420465 p^{9} T^{24} + 26862 p^{10} T^{25} + 31197 p^{11} T^{26} + 81 p^{12} T^{27} + 255 p^{13} T^{28} + p^{15} T^{30} \) | |
| 37 | \( 1 - 12 T + 264 T^{2} - 2227 T^{3} + 29055 T^{4} - 200874 T^{5} + 2103495 T^{6} - 13739163 T^{7} + 126136374 T^{8} - 813583679 T^{9} + 6493964829 T^{10} - 41027041986 T^{11} + 290820826046 T^{12} - 1816234188282 T^{13} + 11851930431117 T^{14} - 71700959122011 T^{15} + 11851930431117 p T^{16} - 1816234188282 p^{2} T^{17} + 290820826046 p^{3} T^{18} - 41027041986 p^{4} T^{19} + 6493964829 p^{5} T^{20} - 813583679 p^{6} T^{21} + 126136374 p^{7} T^{22} - 13739163 p^{8} T^{23} + 2103495 p^{9} T^{24} - 200874 p^{10} T^{25} + 29055 p^{11} T^{26} - 2227 p^{12} T^{27} + 264 p^{13} T^{28} - 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 18 T + 423 T^{2} + 5027 T^{3} + 72180 T^{4} + 682713 T^{5} + 7781013 T^{6} + 64315869 T^{7} + 637749918 T^{8} + 4791228692 T^{9} + 42653046549 T^{10} + 294719836173 T^{11} + 2388615610744 T^{12} + 15262739524797 T^{13} + 113856716947980 T^{14} + 675121974231925 T^{15} + 113856716947980 p T^{16} + 15262739524797 p^{2} T^{17} + 2388615610744 p^{3} T^{18} + 294719836173 p^{4} T^{19} + 42653046549 p^{5} T^{20} + 4791228692 p^{6} T^{21} + 637749918 p^{7} T^{22} + 64315869 p^{8} T^{23} + 7781013 p^{9} T^{24} + 682713 p^{10} T^{25} + 72180 p^{11} T^{26} + 5027 p^{12} T^{27} + 423 p^{13} T^{28} + 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 + 3 T + 264 T^{2} + 609 T^{3} + 36498 T^{4} + 40284 T^{5} + 3369476 T^{6} - 1608552 T^{7} + 228538530 T^{8} - 580173037 T^{9} + 12133077522 T^{10} - 62652467229 T^{11} + 540274019553 T^{12} - 4330119643944 T^{13} + 22394320099728 T^{14} - 216185687948881 T^{15} + 22394320099728 p T^{16} - 4330119643944 p^{2} T^{17} + 540274019553 p^{3} T^{18} - 62652467229 p^{4} T^{19} + 12133077522 p^{5} T^{20} - 580173037 p^{6} T^{21} + 228538530 p^{7} T^{22} - 1608552 p^{8} T^{23} + 3369476 p^{9} T^{24} + 40284 p^{10} T^{25} + 36498 p^{11} T^{26} + 609 p^{12} T^{27} + 264 p^{13} T^{28} + 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + 3 T + 402 T^{2} + 601 T^{3} + 80109 T^{4} + 26829 T^{5} + 10709577 T^{6} - 5592432 T^{7} + 1082502165 T^{8} - 1188848464 T^{9} + 87585238083 T^{10} - 2663789133 p T^{11} + 5838496519672 T^{12} - 9076234922688 T^{13} + 325624475183277 T^{14} - 489927016920204 T^{15} + 325624475183277 p T^{16} - 9076234922688 p^{2} T^{17} + 5838496519672 p^{3} T^{18} - 2663789133 p^{5} T^{19} + 87585238083 p^{5} T^{20} - 1188848464 p^{6} T^{21} + 1082502165 p^{7} T^{22} - 5592432 p^{8} T^{23} + 10709577 p^{9} T^{24} + 26829 p^{10} T^{25} + 80109 p^{11} T^{26} + 601 p^{12} T^{27} + 402 p^{13} T^{28} + 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 + 366 T^{2} + 97 T^{3} + 69627 T^{4} + 39543 T^{5} + 9009920 T^{6} + 8757972 T^{7} + 886410030 T^{8} + 1273610745 T^{9} + 70676111433 T^{10} + 133006243590 T^{11} + 4779306270052 T^{12} + 10430184454077 T^{13} + 283427098714410 T^{14} + 628567103920729 T^{15} + 283427098714410 p T^{16} + 10430184454077 p^{2} T^{17} + 4779306270052 p^{3} T^{18} + 133006243590 p^{4} T^{19} + 70676111433 p^{5} T^{20} + 1273610745 p^{6} T^{21} + 886410030 p^{7} T^{22} + 8757972 p^{8} T^{23} + 9009920 p^{9} T^{24} + 39543 p^{10} T^{25} + 69627 p^{11} T^{26} + 97 p^{12} T^{27} + 366 p^{13} T^{28} + p^{15} T^{30} \) | |
| 59 | \( 1 + 12 T + 582 T^{2} + 5665 T^{3} + 160785 T^{4} + 1335651 T^{5} + 28883876 T^{6} + 211333809 T^{7} + 3831021261 T^{8} + 25154324571 T^{9} + 399649364829 T^{10} + 2381187060339 T^{11} + 33947627754835 T^{12} + 184724727548787 T^{13} + 2393176168225425 T^{14} + 11922021976749829 T^{15} + 2393176168225425 p T^{16} + 184724727548787 p^{2} T^{17} + 33947627754835 p^{3} T^{18} + 2381187060339 p^{4} T^{19} + 399649364829 p^{5} T^{20} + 25154324571 p^{6} T^{21} + 3831021261 p^{7} T^{22} + 211333809 p^{8} T^{23} + 28883876 p^{9} T^{24} + 1335651 p^{10} T^{25} + 160785 p^{11} T^{26} + 5665 p^{12} T^{27} + 582 p^{13} T^{28} + 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 15 T + 435 T^{2} + 5571 T^{3} + 88971 T^{4} + 960372 T^{5} + 11562293 T^{6} + 107060565 T^{7} + 1115266593 T^{8} + 153442078 p T^{9} + 91037625351 T^{10} + 734925209046 T^{11} + 6871060881879 T^{12} + 53773548938049 T^{13} + 477233637971859 T^{14} + 3517937516414113 T^{15} + 477233637971859 p T^{16} + 53773548938049 p^{2} T^{17} + 6871060881879 p^{3} T^{18} + 734925209046 p^{4} T^{19} + 91037625351 p^{5} T^{20} + 153442078 p^{7} T^{21} + 1115266593 p^{7} T^{22} + 107060565 p^{8} T^{23} + 11562293 p^{9} T^{24} + 960372 p^{10} T^{25} + 88971 p^{11} T^{26} + 5571 p^{12} T^{27} + 435 p^{13} T^{28} + 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 + 24 T + 840 T^{2} + 14034 T^{3} + 293142 T^{4} + 3875763 T^{5} + 61787889 T^{6} + 688289700 T^{7} + 9191158302 T^{8} + 89427825106 T^{9} + 1050259116636 T^{10} + 9148885274637 T^{11} + 97380527079886 T^{12} + 773989907511441 T^{13} + 7601063478168855 T^{14} + 55846628345439765 T^{15} + 7601063478168855 p T^{16} + 773989907511441 p^{2} T^{17} + 97380527079886 p^{3} T^{18} + 9148885274637 p^{4} T^{19} + 1050259116636 p^{5} T^{20} + 89427825106 p^{6} T^{21} + 9191158302 p^{7} T^{22} + 688289700 p^{8} T^{23} + 61787889 p^{9} T^{24} + 3875763 p^{10} T^{25} + 293142 p^{11} T^{26} + 14034 p^{12} T^{27} + 840 p^{13} T^{28} + 24 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 - 6 T + 573 T^{2} - 2249 T^{3} + 153609 T^{4} - 326307 T^{5} + 26792710 T^{6} - 17881377 T^{7} + 3569862687 T^{8} + 1245705668 T^{9} + 394417345272 T^{10} + 375237362310 T^{11} + 37211523778396 T^{12} + 46535877522378 T^{13} + 3031799537740167 T^{14} + 55141338474135 p T^{15} + 3031799537740167 p T^{16} + 46535877522378 p^{2} T^{17} + 37211523778396 p^{3} T^{18} + 375237362310 p^{4} T^{19} + 394417345272 p^{5} T^{20} + 1245705668 p^{6} T^{21} + 3569862687 p^{7} T^{22} - 17881377 p^{8} T^{23} + 26792710 p^{9} T^{24} - 326307 p^{10} T^{25} + 153609 p^{11} T^{26} - 2249 p^{12} T^{27} + 573 p^{13} T^{28} - 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 9 T + 534 T^{2} + 5620 T^{3} + 151071 T^{4} + 1692528 T^{5} + 30037891 T^{6} + 334474647 T^{7} + 4645300563 T^{8} + 49112993407 T^{9} + 583375388892 T^{10} + 5711645763189 T^{11} + 60670980060268 T^{12} + 544816527619020 T^{13} + 5275541085051711 T^{14} + 43398767074631933 T^{15} + 5275541085051711 p T^{16} + 544816527619020 p^{2} T^{17} + 60670980060268 p^{3} T^{18} + 5711645763189 p^{4} T^{19} + 583375388892 p^{5} T^{20} + 49112993407 p^{6} T^{21} + 4645300563 p^{7} T^{22} + 334474647 p^{8} T^{23} + 30037891 p^{9} T^{24} + 1692528 p^{10} T^{25} + 151071 p^{11} T^{26} + 5620 p^{12} T^{27} + 534 p^{13} T^{28} + 9 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 + 9 T + 702 T^{2} + 6958 T^{3} + 248004 T^{4} + 2601825 T^{5} + 58635640 T^{6} + 626753088 T^{7} + 10387278894 T^{8} + 109248195268 T^{9} + 1458343325610 T^{10} + 14662443892293 T^{11} + 2114915118244 p T^{12} + 1571195357694957 T^{13} + 15848249650817532 T^{14} + 137110342949221517 T^{15} + 15848249650817532 p T^{16} + 1571195357694957 p^{2} T^{17} + 2114915118244 p^{4} T^{18} + 14662443892293 p^{4} T^{19} + 1458343325610 p^{5} T^{20} + 109248195268 p^{6} T^{21} + 10387278894 p^{7} T^{22} + 626753088 p^{8} T^{23} + 58635640 p^{9} T^{24} + 2601825 p^{10} T^{25} + 248004 p^{11} T^{26} + 6958 p^{12} T^{27} + 702 p^{13} T^{28} + 9 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 - 15 T + 867 T^{2} - 12180 T^{3} + 375633 T^{4} - 4865313 T^{5} + 106657112 T^{6} - 1266051624 T^{7} + 22061140161 T^{8} - 239449383764 T^{9} + 3510503011845 T^{10} - 34779306029907 T^{11} + 443301216529375 T^{12} - 3997687940742780 T^{13} + 45204557892970905 T^{14} - 369264557357643521 T^{15} + 45204557892970905 p T^{16} - 3997687940742780 p^{2} T^{17} + 443301216529375 p^{3} T^{18} - 34779306029907 p^{4} T^{19} + 3510503011845 p^{5} T^{20} - 239449383764 p^{6} T^{21} + 22061140161 p^{7} T^{22} - 1266051624 p^{8} T^{23} + 106657112 p^{9} T^{24} - 4865313 p^{10} T^{25} + 375633 p^{11} T^{26} - 12180 p^{12} T^{27} + 867 p^{13} T^{28} - 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 24 T + 963 T^{2} + 17983 T^{3} + 427857 T^{4} + 6693753 T^{5} + 120906942 T^{6} + 1650271755 T^{7} + 24791393859 T^{8} + 302346084740 T^{9} + 3951603433404 T^{10} + 43666536495114 T^{11} + 509235642695200 T^{12} + 5137131077123676 T^{13} + 54253675263851253 T^{14} + 500855471778889473 T^{15} + 54253675263851253 p T^{16} + 5137131077123676 p^{2} T^{17} + 509235642695200 p^{3} T^{18} + 43666536495114 p^{4} T^{19} + 3951603433404 p^{5} T^{20} + 302346084740 p^{6} T^{21} + 24791393859 p^{7} T^{22} + 1650271755 p^{8} T^{23} + 120906942 p^{9} T^{24} + 6693753 p^{10} T^{25} + 427857 p^{11} T^{26} + 17983 p^{12} T^{27} + 963 p^{13} T^{28} + 24 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 - 48 T + 1977 T^{2} - 55498 T^{3} + 1398888 T^{4} - 29098053 T^{5} + 558608203 T^{6} - 9444087276 T^{7} + 149816919570 T^{8} - 2158305354605 T^{9} + 29480589700137 T^{10} - 371833983056016 T^{11} + 4475974764066578 T^{12} - 50212228301072877 T^{13} + 539595538904021304 T^{14} - 5426545392081710923 T^{15} + 539595538904021304 p T^{16} - 50212228301072877 p^{2} T^{17} + 4475974764066578 p^{3} T^{18} - 371833983056016 p^{4} T^{19} + 29480589700137 p^{5} T^{20} - 2158305354605 p^{6} T^{21} + 149816919570 p^{7} T^{22} - 9444087276 p^{8} T^{23} + 558608203 p^{9} T^{24} - 29098053 p^{10} T^{25} + 1398888 p^{11} T^{26} - 55498 p^{12} T^{27} + 1977 p^{13} T^{28} - 48 p^{14} T^{29} + p^{15} T^{30} \) | |
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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
−1.92297645420339463745157134279, −1.78996295334555753138915200501, −1.73678782068863249227279157796, −1.67727477703597286778176146151, −1.64669920210654910019782423623, −1.62911437198766291868604737849, −1.59323645293170616129981210137, −1.57590035984357710839687598517, −1.54070944188139515583873394973, −1.50022688062430377686705025351, −1.38293376869374210501249033513, −1.15312957064579406197223828476, −1.13670409659300194108308630592, −1.13078409754800116510139881444, −0.993101471901926181675802169792, −0.850118927867107325193975638780, −0.75678121820611436084272475191, −0.73413163314516577885381582160, −0.62951262991853890807379669281, −0.58776835990310961820231689416, −0.50728709301937332696115528690, −0.30986850100399899259004208768, −0.29897439390290666382164845093, −0.26891196719410154404863182842, −0.18328634700497523562502541565, 0.18328634700497523562502541565, 0.26891196719410154404863182842, 0.29897439390290666382164845093, 0.30986850100399899259004208768, 0.50728709301937332696115528690, 0.58776835990310961820231689416, 0.62951262991853890807379669281, 0.73413163314516577885381582160, 0.75678121820611436084272475191, 0.850118927867107325193975638780, 0.993101471901926181675802169792, 1.13078409754800116510139881444, 1.13670409659300194108308630592, 1.15312957064579406197223828476, 1.38293376869374210501249033513, 1.50022688062430377686705025351, 1.54070944188139515583873394973, 1.57590035984357710839687598517, 1.59323645293170616129981210137, 1.62911437198766291868604737849, 1.64669920210654910019782423623, 1.67727477703597286778176146151, 1.73678782068863249227279157796, 1.78996295334555753138915200501, 1.92297645420339463745157134279
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.