Dirichlet series
| L(s) = 1 | − 4·3-s + 8·5-s − 10·7-s + 8·9-s + 16·11-s + 72·13-s − 32·15-s − 40·17-s + 40·21-s + 246·23-s + 32·25-s + 144·27-s + 260·29-s + 566·31-s − 64·33-s − 80·35-s − 28·37-s − 288·39-s − 786·41-s + 64·45-s + 448·47-s + 50·49-s + 160·51-s + 128·55-s + 660·61-s − 80·63-s + 576·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.715·5-s − 0.539·7-s + 8/27·9-s + 0.438·11-s + 1.53·13-s − 0.550·15-s − 0.570·17-s + 0.415·21-s + 2.23·23-s + 0.255·25-s + 1.02·27-s + 1.66·29-s + 3.27·31-s − 0.337·33-s − 0.386·35-s − 0.124·37-s − 1.18·39-s − 2.99·41-s + 0.212·45-s + 1.39·47-s + 0.145·49-s + 0.439·51-s + 0.313·55-s + 1.38·61-s − 0.159·63-s + 1.09·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 17^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 17^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42} \cdot 17^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.58848\times 10^{12}\) |
| Root analytic conductor: | \(2.83271\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} \cdot 17^{14} ,\ ( \ : [3/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1036304758\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1036304758\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 40 T - 15013 T^{2} - 846832 T^{3} + 5137129 p T^{4} + 24811608 p^{2} T^{5} - 36890693 p^{3} T^{6} - 473547040 p^{4} T^{7} - 36890693 p^{6} T^{8} + 24811608 p^{8} T^{9} + 5137129 p^{10} T^{10} - 846832 p^{12} T^{11} - 15013 p^{15} T^{12} + 40 p^{18} T^{13} + p^{21} T^{14} \) | |
| good | 3 | \( 1 + 4 T + 8 T^{2} - 16 p^{2} T^{3} - 551 p T^{4} - 2468 p T^{5} - 2008 p T^{6} + 5500 p^{2} T^{7} + 305659 p T^{8} + 2034520 p T^{9} + 9134120 p T^{10} + 18917804 p^{2} T^{11} - 177374573 T^{12} - 5580375656 T^{13} - 38565642736 T^{14} - 5580375656 p^{3} T^{15} - 177374573 p^{6} T^{16} + 18917804 p^{11} T^{17} + 9134120 p^{13} T^{18} + 2034520 p^{16} T^{19} + 305659 p^{19} T^{20} + 5500 p^{23} T^{21} - 2008 p^{25} T^{22} - 2468 p^{28} T^{23} - 551 p^{31} T^{24} - 16 p^{35} T^{25} + 8 p^{36} T^{26} + 4 p^{39} T^{27} + p^{42} T^{28} \) |
| 5 | \( 1 - 8 T + 32 T^{2} + 48 p^{2} T^{3} - 15133 T^{4} + 24056 p T^{5} + 242016 T^{6} + 1335576 T^{7} + 86204713 T^{8} + 1637794912 T^{9} - 3362151584 T^{10} + 210017702744 T^{11} + 1321460104419 T^{12} - 23811941273424 T^{13} + 526373544037952 T^{14} - 23811941273424 p^{3} T^{15} + 1321460104419 p^{6} T^{16} + 210017702744 p^{9} T^{17} - 3362151584 p^{12} T^{18} + 1637794912 p^{15} T^{19} + 86204713 p^{18} T^{20} + 1335576 p^{21} T^{21} + 242016 p^{24} T^{22} + 24056 p^{28} T^{23} - 15133 p^{30} T^{24} + 48 p^{35} T^{25} + 32 p^{36} T^{26} - 8 p^{39} T^{27} + p^{42} T^{28} \) | |
| 7 | \( 1 + 10 T + 50 T^{2} - 4142 T^{3} - 317137 T^{4} + 414984 T^{5} + 28584772 T^{6} + 2263478104 T^{7} + 46750800541 T^{8} - 646739332406 T^{9} - 9028864959986 T^{10} - 413627427015646 T^{11} - 3251419515 p^{7} T^{12} + 52591829970320 p^{4} T^{13} + 1203390796795775352 T^{14} + 52591829970320 p^{7} T^{15} - 3251419515 p^{13} T^{16} - 413627427015646 p^{9} T^{17} - 9028864959986 p^{12} T^{18} - 646739332406 p^{15} T^{19} + 46750800541 p^{18} T^{20} + 2263478104 p^{21} T^{21} + 28584772 p^{24} T^{22} + 414984 p^{27} T^{23} - 317137 p^{30} T^{24} - 4142 p^{33} T^{25} + 50 p^{36} T^{26} + 10 p^{39} T^{27} + p^{42} T^{28} \) | |
| 11 | \( 1 - 16 T + 128 T^{2} + 76020 T^{3} - 2343141 T^{4} - 5828316 p T^{5} + 4215225864 T^{6} - 137761354972 T^{7} - 4778244141263 T^{8} + 240449321023580 T^{9} - 1776182867618944 T^{10} - 277176617906385496 T^{11} + 10275425936006076323 T^{12} + 20611375648059780600 p T^{13} - \)\(13\!\cdots\!44\)\( p^{2} T^{14} + 20611375648059780600 p^{4} T^{15} + 10275425936006076323 p^{6} T^{16} - 277176617906385496 p^{9} T^{17} - 1776182867618944 p^{12} T^{18} + 240449321023580 p^{15} T^{19} - 4778244141263 p^{18} T^{20} - 137761354972 p^{21} T^{21} + 4215225864 p^{24} T^{22} - 5828316 p^{28} T^{23} - 2343141 p^{30} T^{24} + 76020 p^{33} T^{25} + 128 p^{36} T^{26} - 16 p^{39} T^{27} + p^{42} T^{28} \) | |
| 13 | \( ( 1 - 36 T + 6503 T^{2} - 343680 T^{3} + 24176193 T^{4} - 105759468 p T^{5} + 74002731535 T^{6} - 3489293271296 T^{7} + 74002731535 p^{3} T^{8} - 105759468 p^{7} T^{9} + 24176193 p^{9} T^{10} - 343680 p^{12} T^{11} + 6503 p^{15} T^{12} - 36 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 19 | \( 1 - 28390 T^{2} + 353508403 T^{4} - 2461476201500 T^{6} + 10267900687412745 T^{8} - 27418277564459556058 T^{10} + \)\(73\!\cdots\!87\)\( T^{12} - \)\(39\!\cdots\!28\)\( T^{14} + \)\(73\!\cdots\!87\)\( p^{6} T^{16} - 27418277564459556058 p^{12} T^{18} + 10267900687412745 p^{18} T^{20} - 2461476201500 p^{24} T^{22} + 353508403 p^{30} T^{24} - 28390 p^{36} T^{26} + p^{42} T^{28} \) | |
| 23 | \( 1 - 246 T + 30258 T^{2} - 3323894 T^{3} + 553650287 T^{4} - 85729302688 T^{5} + 9861193738820 T^{6} - 1030016160555712 T^{7} + 110771993354477501 T^{8} - 556221704943734306 p T^{9} + \)\(14\!\cdots\!50\)\( T^{10} - \)\(15\!\cdots\!78\)\( T^{11} + \)\(15\!\cdots\!15\)\( T^{12} - \)\(16\!\cdots\!36\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} - \)\(16\!\cdots\!36\)\( p^{3} T^{15} + \)\(15\!\cdots\!15\)\( p^{6} T^{16} - \)\(15\!\cdots\!78\)\( p^{9} T^{17} + \)\(14\!\cdots\!50\)\( p^{12} T^{18} - 556221704943734306 p^{16} T^{19} + 110771993354477501 p^{18} T^{20} - 1030016160555712 p^{21} T^{21} + 9861193738820 p^{24} T^{22} - 85729302688 p^{27} T^{23} + 553650287 p^{30} T^{24} - 3323894 p^{33} T^{25} + 30258 p^{36} T^{26} - 246 p^{39} T^{27} + p^{42} T^{28} \) | |
| 29 | \( 1 - 260 T + 33800 T^{2} - 637156 T^{3} - 1236561213 T^{4} + 325671406632 T^{5} - 42675812840752 T^{6} + 2610268324637192 T^{7} + 244334023388148665 T^{8} - 64079110524157156428 T^{9} + \)\(69\!\cdots\!64\)\( T^{10} - \)\(69\!\cdots\!60\)\( T^{11} + \)\(84\!\cdots\!03\)\( T^{12} - \)\(35\!\cdots\!36\)\( T^{13} + \)\(84\!\cdots\!68\)\( T^{14} - \)\(35\!\cdots\!36\)\( p^{3} T^{15} + \)\(84\!\cdots\!03\)\( p^{6} T^{16} - \)\(69\!\cdots\!60\)\( p^{9} T^{17} + \)\(69\!\cdots\!64\)\( p^{12} T^{18} - 64079110524157156428 p^{15} T^{19} + 244334023388148665 p^{18} T^{20} + 2610268324637192 p^{21} T^{21} - 42675812840752 p^{24} T^{22} + 325671406632 p^{27} T^{23} - 1236561213 p^{30} T^{24} - 637156 p^{33} T^{25} + 33800 p^{36} T^{26} - 260 p^{39} T^{27} + p^{42} T^{28} \) | |
| 31 | \( 1 - 566 T + 160178 T^{2} - 44755254 T^{3} + 392291313 p T^{4} - 2528829319456 T^{5} + 484904199169220 T^{6} - 100334924350465664 T^{7} + 17276264919132021293 T^{8} - \)\(23\!\cdots\!58\)\( T^{9} + \)\(38\!\cdots\!38\)\( T^{10} - \)\(54\!\cdots\!30\)\( T^{11} + \)\(46\!\cdots\!99\)\( T^{12} - \)\(58\!\cdots\!36\)\( T^{13} + \)\(14\!\cdots\!96\)\( T^{14} - \)\(58\!\cdots\!36\)\( p^{3} T^{15} + \)\(46\!\cdots\!99\)\( p^{6} T^{16} - \)\(54\!\cdots\!30\)\( p^{9} T^{17} + \)\(38\!\cdots\!38\)\( p^{12} T^{18} - \)\(23\!\cdots\!58\)\( p^{15} T^{19} + 17276264919132021293 p^{18} T^{20} - 100334924350465664 p^{21} T^{21} + 484904199169220 p^{24} T^{22} - 2528829319456 p^{27} T^{23} + 392291313 p^{31} T^{24} - 44755254 p^{33} T^{25} + 160178 p^{36} T^{26} - 566 p^{39} T^{27} + p^{42} T^{28} \) | |
| 37 | \( 1 + 28 T + 392 T^{2} + 7958436 T^{3} + 6314800675 T^{4} - 414936890656 T^{5} + 17574716980080 T^{6} + 32943627166380672 T^{7} + 12179202270523412649 T^{8} - \)\(35\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(27\!\cdots\!84\)\( T^{11} - \)\(66\!\cdots\!49\)\( T^{12} - \)\(94\!\cdots\!40\)\( T^{13} + \)\(47\!\cdots\!96\)\( T^{14} - \)\(94\!\cdots\!40\)\( p^{3} T^{15} - \)\(66\!\cdots\!49\)\( p^{6} T^{16} - \)\(27\!\cdots\!84\)\( p^{9} T^{17} + \)\(13\!\cdots\!72\)\( p^{12} T^{18} - \)\(35\!\cdots\!60\)\( p^{15} T^{19} + 12179202270523412649 p^{18} T^{20} + 32943627166380672 p^{21} T^{21} + 17574716980080 p^{24} T^{22} - 414936890656 p^{27} T^{23} + 6314800675 p^{30} T^{24} + 7958436 p^{33} T^{25} + 392 p^{36} T^{26} + 28 p^{39} T^{27} + p^{42} T^{28} \) | |
| 41 | \( 1 + 786 T + 308898 T^{2} + 89332290 T^{3} + 26768086539 T^{4} + 7149581419796 T^{5} + 1341091618557684 T^{6} + 59368518180264052 T^{7} - 41251674003861349655 T^{8} - \)\(14\!\cdots\!98\)\( T^{9} - \)\(33\!\cdots\!26\)\( T^{10} - \)\(99\!\cdots\!74\)\( T^{11} - \)\(10\!\cdots\!49\)\( T^{12} + \)\(37\!\cdots\!08\)\( T^{13} + \)\(19\!\cdots\!08\)\( T^{14} + \)\(37\!\cdots\!08\)\( p^{3} T^{15} - \)\(10\!\cdots\!49\)\( p^{6} T^{16} - \)\(99\!\cdots\!74\)\( p^{9} T^{17} - \)\(33\!\cdots\!26\)\( p^{12} T^{18} - \)\(14\!\cdots\!98\)\( p^{15} T^{19} - 41251674003861349655 p^{18} T^{20} + 59368518180264052 p^{21} T^{21} + 1341091618557684 p^{24} T^{22} + 7149581419796 p^{27} T^{23} + 26768086539 p^{30} T^{24} + 89332290 p^{33} T^{25} + 308898 p^{36} T^{26} + 786 p^{39} T^{27} + p^{42} T^{28} \) | |
| 43 | \( 1 - 686098 T^{2} + 239848417539 T^{4} - 55925021126130132 T^{6} + \)\(96\!\cdots\!53\)\( T^{8} - \)\(12\!\cdots\!42\)\( T^{10} + \)\(14\!\cdots\!19\)\( T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!19\)\( p^{6} T^{16} - \)\(12\!\cdots\!42\)\( p^{12} T^{18} + \)\(96\!\cdots\!53\)\( p^{18} T^{20} - 55925021126130132 p^{24} T^{22} + 239848417539 p^{30} T^{24} - 686098 p^{36} T^{26} + p^{42} T^{28} \) | |
| 47 | \( ( 1 - 224 T + 462777 T^{2} - 67046720 T^{3} + 106497238117 T^{4} - 12223536344864 T^{5} + 16129117638991629 T^{6} - 1528856107438378368 T^{7} + 16129117638991629 p^{3} T^{8} - 12223536344864 p^{6} T^{9} + 106497238117 p^{9} T^{10} - 67046720 p^{12} T^{11} + 462777 p^{15} T^{12} - 224 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 53 | \( 1 - 1559298 T^{2} + 1182242543139 T^{4} - 10884389699555428 p T^{6} + \)\(20\!\cdots\!09\)\( T^{8} - \)\(53\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!99\)\( T^{12} - \)\(18\!\cdots\!92\)\( T^{14} + \)\(11\!\cdots\!99\)\( p^{6} T^{16} - \)\(53\!\cdots\!54\)\( p^{12} T^{18} + \)\(20\!\cdots\!09\)\( p^{18} T^{20} - 10884389699555428 p^{25} T^{22} + 1182242543139 p^{30} T^{24} - 1559298 p^{36} T^{26} + p^{42} T^{28} \) | |
| 59 | \( 1 - 2001378 T^{2} + 1965959044707 T^{4} - 1257971452112238708 T^{6} + \)\(58\!\cdots\!21\)\( T^{8} - \)\(20\!\cdots\!66\)\( T^{10} + \)\(59\!\cdots\!43\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{14} + \)\(59\!\cdots\!43\)\( p^{6} T^{16} - \)\(20\!\cdots\!66\)\( p^{12} T^{18} + \)\(58\!\cdots\!21\)\( p^{18} T^{20} - 1257971452112238708 p^{24} T^{22} + 1965959044707 p^{30} T^{24} - 2001378 p^{36} T^{26} + p^{42} T^{28} \) | |
| 61 | \( 1 - 660 T + 217800 T^{2} - 246269116 T^{3} + 70682222291 T^{4} + 46810598867664 T^{5} - 15965344519927312 T^{6} + 18402895702331966064 T^{7} - \)\(18\!\cdots\!39\)\( T^{8} + \)\(41\!\cdots\!04\)\( T^{9} - \)\(93\!\cdots\!24\)\( T^{10} + \)\(71\!\cdots\!08\)\( T^{11} + \)\(67\!\cdots\!83\)\( T^{12} - \)\(47\!\cdots\!44\)\( T^{13} + \)\(12\!\cdots\!32\)\( T^{14} - \)\(47\!\cdots\!44\)\( p^{3} T^{15} + \)\(67\!\cdots\!83\)\( p^{6} T^{16} + \)\(71\!\cdots\!08\)\( p^{9} T^{17} - \)\(93\!\cdots\!24\)\( p^{12} T^{18} + \)\(41\!\cdots\!04\)\( p^{15} T^{19} - \)\(18\!\cdots\!39\)\( p^{18} T^{20} + 18402895702331966064 p^{21} T^{21} - 15965344519927312 p^{24} T^{22} + 46810598867664 p^{27} T^{23} + 70682222291 p^{30} T^{24} - 246269116 p^{33} T^{25} + 217800 p^{36} T^{26} - 660 p^{39} T^{27} + p^{42} T^{28} \) | |
| 67 | \( ( 1 + 142 T + 1486133 T^{2} + 264377148 T^{3} + 1024540731093 T^{4} + 201170030068994 T^{5} + 441913797793890425 T^{6} + 80891859699015339144 T^{7} + 441913797793890425 p^{3} T^{8} + 201170030068994 p^{6} T^{9} + 1024540731093 p^{9} T^{10} + 264377148 p^{12} T^{11} + 1486133 p^{15} T^{12} + 142 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 71 | \( 1 - 326 T + 53138 T^{2} + 95557586 T^{3} + 55342648767 T^{4} - 125581174913560 T^{5} + 42564291472713412 T^{6} - 30208580535548686792 T^{7} - \)\(16\!\cdots\!71\)\( T^{8} + \)\(11\!\cdots\!02\)\( T^{9} + \)\(31\!\cdots\!30\)\( T^{10} - \)\(63\!\cdots\!66\)\( T^{11} + \)\(42\!\cdots\!83\)\( T^{12} + \)\(10\!\cdots\!56\)\( T^{13} - \)\(32\!\cdots\!32\)\( T^{14} + \)\(10\!\cdots\!56\)\( p^{3} T^{15} + \)\(42\!\cdots\!83\)\( p^{6} T^{16} - \)\(63\!\cdots\!66\)\( p^{9} T^{17} + \)\(31\!\cdots\!30\)\( p^{12} T^{18} + \)\(11\!\cdots\!02\)\( p^{15} T^{19} - \)\(16\!\cdots\!71\)\( p^{18} T^{20} - 30208580535548686792 p^{21} T^{21} + 42564291472713412 p^{24} T^{22} - 125581174913560 p^{27} T^{23} + 55342648767 p^{30} T^{24} + 95557586 p^{33} T^{25} + 53138 p^{36} T^{26} - 326 p^{39} T^{27} + p^{42} T^{28} \) | |
| 73 | \( 1 - 10 p T + 50 p^{2} T^{2} - 286924138 T^{3} - 159353054069 T^{4} + 215969782490364 T^{5} - 74035589477759148 T^{6} + 76215878493591509788 T^{7} - \)\(34\!\cdots\!63\)\( T^{8} - \)\(27\!\cdots\!30\)\( T^{9} - \)\(91\!\cdots\!58\)\( T^{10} + \)\(19\!\cdots\!78\)\( T^{11} + \)\(55\!\cdots\!63\)\( T^{12} - \)\(51\!\cdots\!20\)\( T^{13} + \)\(18\!\cdots\!12\)\( T^{14} - \)\(51\!\cdots\!20\)\( p^{3} T^{15} + \)\(55\!\cdots\!63\)\( p^{6} T^{16} + \)\(19\!\cdots\!78\)\( p^{9} T^{17} - \)\(91\!\cdots\!58\)\( p^{12} T^{18} - \)\(27\!\cdots\!30\)\( p^{15} T^{19} - \)\(34\!\cdots\!63\)\( p^{18} T^{20} + 76215878493591509788 p^{21} T^{21} - 74035589477759148 p^{24} T^{22} + 215969782490364 p^{27} T^{23} - 159353054069 p^{30} T^{24} - 286924138 p^{33} T^{25} + 50 p^{38} T^{26} - 10 p^{40} T^{27} + p^{42} T^{28} \) | |
| 79 | \( 1 - 342 T + 58482 T^{2} - 33137334 T^{3} + 335890070463 T^{4} - 171058359363280 T^{5} + 39407477253738372 T^{6} - 47228136621680925840 T^{7} + \)\(37\!\cdots\!65\)\( T^{8} - \)\(24\!\cdots\!74\)\( T^{9} + \)\(90\!\cdots\!02\)\( T^{10} - \)\(16\!\cdots\!02\)\( T^{11} + \)\(21\!\cdots\!87\)\( T^{12} - \)\(42\!\cdots\!80\)\( T^{13} + \)\(14\!\cdots\!08\)\( T^{14} - \)\(42\!\cdots\!80\)\( p^{3} T^{15} + \)\(21\!\cdots\!87\)\( p^{6} T^{16} - \)\(16\!\cdots\!02\)\( p^{9} T^{17} + \)\(90\!\cdots\!02\)\( p^{12} T^{18} - \)\(24\!\cdots\!74\)\( p^{15} T^{19} + \)\(37\!\cdots\!65\)\( p^{18} T^{20} - 47228136621680925840 p^{21} T^{21} + 39407477253738372 p^{24} T^{22} - 171058359363280 p^{27} T^{23} + 335890070463 p^{30} T^{24} - 33137334 p^{33} T^{25} + 58482 p^{36} T^{26} - 342 p^{39} T^{27} + p^{42} T^{28} \) | |
| 83 | \( 1 - 3566066 T^{2} + 6336753165107 T^{4} - 7647889272777785300 T^{6} + \)\(72\!\cdots\!65\)\( T^{8} - \)\(58\!\cdots\!82\)\( T^{10} + \)\(40\!\cdots\!11\)\( T^{12} - \)\(25\!\cdots\!28\)\( T^{14} + \)\(40\!\cdots\!11\)\( p^{6} T^{16} - \)\(58\!\cdots\!82\)\( p^{12} T^{18} + \)\(72\!\cdots\!65\)\( p^{18} T^{20} - 7647889272777785300 p^{24} T^{22} + 6336753165107 p^{30} T^{24} - 3566066 p^{36} T^{26} + p^{42} T^{28} \) | |
| 89 | \( ( 1 + 2258 T + 3734687 T^{2} + 3763668948 T^{3} + 2741175629205 T^{4} + 916503435920334 T^{5} - 341025530971115621 T^{6} - \)\(76\!\cdots\!16\)\( T^{7} - 341025530971115621 p^{3} T^{8} + 916503435920334 p^{6} T^{9} + 2741175629205 p^{9} T^{10} + 3763668948 p^{12} T^{11} + 3734687 p^{15} T^{12} + 2258 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 97 | \( 1 + 1194 T + 712818 T^{2} - 1319191062 T^{3} - 701429090565 T^{4} + 1358938544055684 T^{5} + 2992696432111192788 T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(23\!\cdots\!39\)\( T^{8} - \)\(28\!\cdots\!86\)\( T^{9} - \)\(26\!\cdots\!94\)\( T^{10} + \)\(30\!\cdots\!98\)\( T^{11} + \)\(19\!\cdots\!91\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{13} - \)\(38\!\cdots\!56\)\( T^{14} - \)\(16\!\cdots\!60\)\( p^{3} T^{15} + \)\(19\!\cdots\!91\)\( p^{6} T^{16} + \)\(30\!\cdots\!98\)\( p^{9} T^{17} - \)\(26\!\cdots\!94\)\( p^{12} T^{18} - \)\(28\!\cdots\!86\)\( p^{15} T^{19} - \)\(23\!\cdots\!39\)\( p^{18} T^{20} + \)\(11\!\cdots\!80\)\( p^{21} T^{21} + 2992696432111192788 p^{24} T^{22} + 1358938544055684 p^{27} T^{23} - 701429090565 p^{30} T^{24} - 1319191062 p^{33} T^{25} + 712818 p^{36} T^{26} + 1194 p^{39} T^{27} + p^{42} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.68303123834572965613592634693, −3.57853190685070924937633171417, −3.54525720427025823482191912807, −3.36880307373346307331405202463, −3.28208904269142924386181510213, −2.97419608193696650146584812316, −2.91913428715303958771719664183, −2.90121017223985845516811407379, −2.82134322448767674448824007081, −2.55539295186224484634760123410, −2.45340848092167292382468265273, −2.35615342941138972975326862534, −2.35511075846946009052339154374, −2.25159096576282137173435757605, −1.81649218036859036917095185151, −1.53154969192464018096912420351, −1.49878591575231416342407811342, −1.47793631987028331238989689027, −1.20480038616695342519747539929, −1.01589817565930965961255522807, −0.980197940736618991093132327547, −0.793452737883932713034680723369, −0.791265901506604944113914760298, −0.23007972716586429078819234140, −0.02626962732747981171969517406, 0.02626962732747981171969517406, 0.23007972716586429078819234140, 0.791265901506604944113914760298, 0.793452737883932713034680723369, 0.980197940736618991093132327547, 1.01589817565930965961255522807, 1.20480038616695342519747539929, 1.47793631987028331238989689027, 1.49878591575231416342407811342, 1.53154969192464018096912420351, 1.81649218036859036917095185151, 2.25159096576282137173435757605, 2.35511075846946009052339154374, 2.35615342941138972975326862534, 2.45340848092167292382468265273, 2.55539295186224484634760123410, 2.82134322448767674448824007081, 2.90121017223985845516811407379, 2.91913428715303958771719664183, 2.97419608193696650146584812316, 3.28208904269142924386181510213, 3.36880307373346307331405202463, 3.54525720427025823482191912807, 3.57853190685070924937633171417, 3.68303123834572965613592634693
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.