Dirichlet series
| L(s) = 1 | + 2-s − 3·4-s − 3·5-s + 13·7-s − 4·8-s − 3·10-s − 8·11-s + 19·13-s + 13·14-s − 16-s + 4·17-s + 19·19-s + 9·20-s − 8·22-s + 16·23-s − 24·25-s + 19·26-s − 39·28-s + 16·29-s + 24·31-s + 7·32-s + 4·34-s − 39·35-s + 26·37-s + 19·38-s + 12·40-s + 15·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s − 1.34·5-s + 4.91·7-s − 1.41·8-s − 0.948·10-s − 2.41·11-s + 5.26·13-s + 3.47·14-s − 1/4·16-s + 0.970·17-s + 4.35·19-s + 2.01·20-s − 1.70·22-s + 3.33·23-s − 4.79·25-s + 3.72·26-s − 7.37·28-s + 2.97·29-s + 4.31·31-s + 1.23·32-s + 0.685·34-s − 6.59·35-s + 4.27·37-s + 3.08·38-s + 1.89·40-s + 2.34·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 23^{16} \cdot 29^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.76832\times 10^{26}\) |
| Root analytic conductor: | \(6.92345\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 23^{16} \cdot 29^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3350.458475\) |
| \(L(\frac12)\) | \(\approx\) | \(3350.458475\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 23 | \( ( 1 - T )^{16} \) | |
| 29 | \( ( 1 - T )^{16} \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - 3 T^{3} + 3 p^{2} T^{4} - 13 T^{5} + 37 T^{6} - 41 T^{7} + 43 p T^{8} - 59 p T^{9} + 231 T^{10} - 19 p^{4} T^{11} + 521 T^{12} - 5 p^{7} T^{13} + 283 p^{2} T^{14} - 43 p^{5} T^{15} + 307 p^{3} T^{16} - 43 p^{6} T^{17} + 283 p^{4} T^{18} - 5 p^{10} T^{19} + 521 p^{4} T^{20} - 19 p^{9} T^{21} + 231 p^{6} T^{22} - 59 p^{8} T^{23} + 43 p^{9} T^{24} - 41 p^{9} T^{25} + 37 p^{10} T^{26} - 13 p^{11} T^{27} + 3 p^{14} T^{28} - 3 p^{13} T^{29} + p^{16} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 + 3 T + 33 T^{2} + 83 T^{3} + 572 T^{4} + 1319 T^{5} + 7104 T^{6} + 15349 T^{7} + 69629 T^{8} + 28444 p T^{9} + 568036 T^{10} + 1101608 T^{11} + 3968776 T^{12} + 1458778 p T^{13} + 4823279 p T^{14} + 41856232 T^{15} + 128600972 T^{16} + 41856232 p T^{17} + 4823279 p^{3} T^{18} + 1458778 p^{4} T^{19} + 3968776 p^{4} T^{20} + 1101608 p^{5} T^{21} + 568036 p^{6} T^{22} + 28444 p^{8} T^{23} + 69629 p^{8} T^{24} + 15349 p^{9} T^{25} + 7104 p^{10} T^{26} + 1319 p^{11} T^{27} + 572 p^{12} T^{28} + 83 p^{13} T^{29} + 33 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 - 13 T + 120 T^{2} - 804 T^{3} + 4551 T^{4} - 21729 T^{5} + 91678 T^{6} - 338046 T^{7} + 1105030 T^{8} - 3108264 T^{9} + 7298782 T^{10} - 12074303 T^{11} + 2434585 T^{12} + 89106998 T^{13} - 474263316 T^{14} + 1720403873 T^{15} - 4979877774 T^{16} + 1720403873 p T^{17} - 474263316 p^{2} T^{18} + 89106998 p^{3} T^{19} + 2434585 p^{4} T^{20} - 12074303 p^{5} T^{21} + 7298782 p^{6} T^{22} - 3108264 p^{7} T^{23} + 1105030 p^{8} T^{24} - 338046 p^{9} T^{25} + 91678 p^{10} T^{26} - 21729 p^{11} T^{27} + 4551 p^{12} T^{28} - 804 p^{13} T^{29} + 120 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 8 T + 124 T^{2} + 819 T^{3} + 7197 T^{4} + 40910 T^{5} + 267126 T^{6} + 1341878 T^{7} + 7228206 T^{8} + 32646945 T^{9} + 152778984 T^{10} + 627549447 T^{11} + 2623058407 T^{12} + 9868098664 T^{13} + 37405543590 T^{14} + 129284753107 T^{15} + 448129229178 T^{16} + 129284753107 p T^{17} + 37405543590 p^{2} T^{18} + 9868098664 p^{3} T^{19} + 2623058407 p^{4} T^{20} + 627549447 p^{5} T^{21} + 152778984 p^{6} T^{22} + 32646945 p^{7} T^{23} + 7228206 p^{8} T^{24} + 1341878 p^{9} T^{25} + 267126 p^{10} T^{26} + 40910 p^{11} T^{27} + 7197 p^{12} T^{28} + 819 p^{13} T^{29} + 124 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 19 T + 298 T^{2} - 3314 T^{3} + 32145 T^{4} - 263999 T^{5} + 1957100 T^{6} - 13012182 T^{7} + 79750453 T^{8} - 450160393 T^{9} + 2373582126 T^{10} - 11690911651 T^{11} + 54236880926 T^{12} - 236892228119 T^{13} + 979270962230 T^{14} - 3826389950627 T^{15} + 14181759836002 T^{16} - 3826389950627 p T^{17} + 979270962230 p^{2} T^{18} - 236892228119 p^{3} T^{19} + 54236880926 p^{4} T^{20} - 11690911651 p^{5} T^{21} + 2373582126 p^{6} T^{22} - 450160393 p^{7} T^{23} + 79750453 p^{8} T^{24} - 13012182 p^{9} T^{25} + 1957100 p^{10} T^{26} - 263999 p^{11} T^{27} + 32145 p^{12} T^{28} - 3314 p^{13} T^{29} + 298 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 4 T + 105 T^{2} - 299 T^{3} + 5458 T^{4} - 12698 T^{5} + 197720 T^{6} - 425660 T^{7} + 5655591 T^{8} - 12646197 T^{9} + 135952466 T^{10} - 334122711 T^{11} + 2846433090 T^{12} - 7686727064 T^{13} + 53717007117 T^{14} - 152110631623 T^{15} + 940947036552 T^{16} - 152110631623 p T^{17} + 53717007117 p^{2} T^{18} - 7686727064 p^{3} T^{19} + 2846433090 p^{4} T^{20} - 334122711 p^{5} T^{21} + 135952466 p^{6} T^{22} - 12646197 p^{7} T^{23} + 5655591 p^{8} T^{24} - 425660 p^{9} T^{25} + 197720 p^{10} T^{26} - 12698 p^{11} T^{27} + 5458 p^{12} T^{28} - 299 p^{13} T^{29} + 105 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - p T + 306 T^{2} - 3175 T^{3} + 29574 T^{4} - 213809 T^{5} + 1447182 T^{6} - 8077449 T^{7} + 44238549 T^{8} - 205703372 T^{9} + 1006667408 T^{10} - 4274869164 T^{11} + 20791502522 T^{12} - 87804548738 T^{13} + 437819221792 T^{14} - 1826900858190 T^{15} + 8801375311540 T^{16} - 1826900858190 p T^{17} + 437819221792 p^{2} T^{18} - 87804548738 p^{3} T^{19} + 20791502522 p^{4} T^{20} - 4274869164 p^{5} T^{21} + 1006667408 p^{6} T^{22} - 205703372 p^{7} T^{23} + 44238549 p^{8} T^{24} - 8077449 p^{9} T^{25} + 1447182 p^{10} T^{26} - 213809 p^{11} T^{27} + 29574 p^{12} T^{28} - 3175 p^{13} T^{29} + 306 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 24 T + 546 T^{2} - 8297 T^{3} + 116585 T^{4} - 1349216 T^{5} + 14569250 T^{6} - 139005886 T^{7} + 1250373298 T^{8} - 10270921065 T^{9} + 80220563558 T^{10} - 582685198433 T^{11} + 4050433584927 T^{12} - 26465173663698 T^{13} + 166223733105942 T^{14} - 987027398318235 T^{15} + 5646756334450186 T^{16} - 987027398318235 p T^{17} + 166223733105942 p^{2} T^{18} - 26465173663698 p^{3} T^{19} + 4050433584927 p^{4} T^{20} - 582685198433 p^{5} T^{21} + 80220563558 p^{6} T^{22} - 10270921065 p^{7} T^{23} + 1250373298 p^{8} T^{24} - 139005886 p^{9} T^{25} + 14569250 p^{10} T^{26} - 1349216 p^{11} T^{27} + 116585 p^{12} T^{28} - 8297 p^{13} T^{29} + 546 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 26 T + 718 T^{2} - 12561 T^{3} + 212768 T^{4} - 2882628 T^{5} + 37271932 T^{6} - 417045923 T^{7} + 4448849589 T^{8} - 42578173809 T^{9} + 389157534608 T^{10} - 3252297755891 T^{11} + 26006216634590 T^{12} - 192148833268307 T^{13} + 1360288879833426 T^{14} - 8945905826461451 T^{15} + 56411283830063528 T^{16} - 8945905826461451 p T^{17} + 1360288879833426 p^{2} T^{18} - 192148833268307 p^{3} T^{19} + 26006216634590 p^{4} T^{20} - 3252297755891 p^{5} T^{21} + 389157534608 p^{6} T^{22} - 42578173809 p^{7} T^{23} + 4448849589 p^{8} T^{24} - 417045923 p^{9} T^{25} + 37271932 p^{10} T^{26} - 2882628 p^{11} T^{27} + 212768 p^{12} T^{28} - 12561 p^{13} T^{29} + 718 p^{14} T^{30} - 26 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 15 T + 427 T^{2} - 4333 T^{3} + 72902 T^{4} - 542103 T^{5} + 7245402 T^{6} - 41306237 T^{7} + 521320635 T^{8} - 2411087708 T^{9} + 31913258818 T^{10} - 127776076240 T^{11} + 1761167235846 T^{12} - 6289776758660 T^{13} + 86170188359457 T^{14} - 279550070438288 T^{15} + 3734275000083968 T^{16} - 279550070438288 p T^{17} + 86170188359457 p^{2} T^{18} - 6289776758660 p^{3} T^{19} + 1761167235846 p^{4} T^{20} - 127776076240 p^{5} T^{21} + 31913258818 p^{6} T^{22} - 2411087708 p^{7} T^{23} + 521320635 p^{8} T^{24} - 41306237 p^{9} T^{25} + 7245402 p^{10} T^{26} - 542103 p^{11} T^{27} + 72902 p^{12} T^{28} - 4333 p^{13} T^{29} + 427 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 33 T + 849 T^{2} - 14973 T^{3} + 227662 T^{4} - 2816959 T^{5} + 31287178 T^{6} - 298052951 T^{7} + 2595602135 T^{8} - 19610166260 T^{9} + 135473232502 T^{10} - 785905431788 T^{11} + 4000938206578 T^{12} - 13776936930866 T^{13} + 20903687462007 T^{14} + 285879832687618 T^{15} - 2444438322463504 T^{16} + 285879832687618 p T^{17} + 20903687462007 p^{2} T^{18} - 13776936930866 p^{3} T^{19} + 4000938206578 p^{4} T^{20} - 785905431788 p^{5} T^{21} + 135473232502 p^{6} T^{22} - 19610166260 p^{7} T^{23} + 2595602135 p^{8} T^{24} - 298052951 p^{9} T^{25} + 31287178 p^{10} T^{26} - 2816959 p^{11} T^{27} + 227662 p^{12} T^{28} - 14973 p^{13} T^{29} + 849 p^{14} T^{30} - 33 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 13 T + 577 T^{2} + 6704 T^{3} + 159740 T^{4} + 1669359 T^{5} + 28317255 T^{6} + 267574444 T^{7} + 3613396300 T^{8} + 31002497946 T^{9} + 353131896133 T^{10} + 2759377979105 T^{11} + 27412240369956 T^{12} + 195352502747250 T^{13} + 1727408754101875 T^{14} + 11218490075789707 T^{15} + 89413246954024486 T^{16} + 11218490075789707 p T^{17} + 1727408754101875 p^{2} T^{18} + 195352502747250 p^{3} T^{19} + 27412240369956 p^{4} T^{20} + 2759377979105 p^{5} T^{21} + 353131896133 p^{6} T^{22} + 31002497946 p^{7} T^{23} + 3613396300 p^{8} T^{24} + 267574444 p^{9} T^{25} + 28317255 p^{10} T^{26} + 1669359 p^{11} T^{27} + 159740 p^{12} T^{28} + 6704 p^{13} T^{29} + 577 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 5 T + 459 T^{2} + 1939 T^{3} + 104759 T^{4} + 409362 T^{5} + 15995137 T^{6} + 62125282 T^{7} + 1837174966 T^{8} + 7385859436 T^{9} + 168847447731 T^{10} + 708468232808 T^{11} + 12909505399193 T^{12} + 55503752465943 T^{13} + 843207959471281 T^{14} + 3572246791815617 T^{15} + 47824741115119442 T^{16} + 3572246791815617 p T^{17} + 843207959471281 p^{2} T^{18} + 55503752465943 p^{3} T^{19} + 12909505399193 p^{4} T^{20} + 708468232808 p^{5} T^{21} + 168847447731 p^{6} T^{22} + 7385859436 p^{7} T^{23} + 1837174966 p^{8} T^{24} + 62125282 p^{9} T^{25} + 15995137 p^{10} T^{26} + 409362 p^{11} T^{27} + 104759 p^{12} T^{28} + 1939 p^{13} T^{29} + 459 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 2 T + 456 T^{2} + 410 T^{3} + 105694 T^{4} + 1578 T^{5} + 16738484 T^{6} - 11214982 T^{7} + 2043859457 T^{8} - 2364599752 T^{9} + 204440098008 T^{10} - 298947595392 T^{11} + 17293348708774 T^{12} - 27860047508768 T^{13} + 1258623500826412 T^{14} - 2043515480003328 T^{15} + 79558310545440212 T^{16} - 2043515480003328 p T^{17} + 1258623500826412 p^{2} T^{18} - 27860047508768 p^{3} T^{19} + 17293348708774 p^{4} T^{20} - 298947595392 p^{5} T^{21} + 204440098008 p^{6} T^{22} - 2364599752 p^{7} T^{23} + 2043859457 p^{8} T^{24} - 11214982 p^{9} T^{25} + 16738484 p^{10} T^{26} + 1578 p^{11} T^{27} + 105694 p^{12} T^{28} + 410 p^{13} T^{29} + 456 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 29 T + 944 T^{2} - 19058 T^{3} + 382210 T^{4} - 6113816 T^{5} + 94573042 T^{6} - 1272502554 T^{7} + 16439619025 T^{8} - 192257946445 T^{9} + 2155532568934 T^{10} - 22340216230423 T^{11} + 222004378224474 T^{12} - 2062961303066295 T^{13} + 18389585981128520 T^{14} - 154188399183324000 T^{15} + 1240580780851833988 T^{16} - 154188399183324000 p T^{17} + 18389585981128520 p^{2} T^{18} - 2062961303066295 p^{3} T^{19} + 222004378224474 p^{4} T^{20} - 22340216230423 p^{5} T^{21} + 2155532568934 p^{6} T^{22} - 192257946445 p^{7} T^{23} + 16439619025 p^{8} T^{24} - 1272502554 p^{9} T^{25} + 94573042 p^{10} T^{26} - 6113816 p^{11} T^{27} + 382210 p^{12} T^{28} - 19058 p^{13} T^{29} + 944 p^{14} T^{30} - 29 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 32 T + 899 T^{2} - 17381 T^{3} + 299650 T^{4} - 4367598 T^{5} + 58180816 T^{6} - 704246868 T^{7} + 7965286745 T^{8} - 84941529999 T^{9} + 863331531798 T^{10} - 8444234612619 T^{11} + 79746261365378 T^{12} - 730107612862990 T^{13} + 6478611978227839 T^{14} - 55688515897313891 T^{15} + 463482632727338148 T^{16} - 55688515897313891 p T^{17} + 6478611978227839 p^{2} T^{18} - 730107612862990 p^{3} T^{19} + 79746261365378 p^{4} T^{20} - 8444234612619 p^{5} T^{21} + 863331531798 p^{6} T^{22} - 84941529999 p^{7} T^{23} + 7965286745 p^{8} T^{24} - 704246868 p^{9} T^{25} + 58180816 p^{10} T^{26} - 4367598 p^{11} T^{27} + 299650 p^{12} T^{28} - 17381 p^{13} T^{29} + 899 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 29 T + 932 T^{2} + 17858 T^{3} + 345833 T^{4} + 5076239 T^{5} + 74234648 T^{6} + 890327184 T^{7} + 10665643953 T^{8} + 108028812469 T^{9} + 1106635368452 T^{10} + 9678586554561 T^{11} + 88074708018966 T^{12} + 687500520363023 T^{13} + 5917159166967580 T^{14} + 44483733398901865 T^{15} + 397289714357614342 T^{16} + 44483733398901865 p T^{17} + 5917159166967580 p^{2} T^{18} + 687500520363023 p^{3} T^{19} + 88074708018966 p^{4} T^{20} + 9678586554561 p^{5} T^{21} + 1106635368452 p^{6} T^{22} + 108028812469 p^{7} T^{23} + 10665643953 p^{8} T^{24} + 890327184 p^{9} T^{25} + 74234648 p^{10} T^{26} + 5076239 p^{11} T^{27} + 345833 p^{12} T^{28} + 17858 p^{13} T^{29} + 932 p^{14} T^{30} + 29 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 19 T + 799 T^{2} - 12074 T^{3} + 294238 T^{4} - 3713937 T^{5} + 67555519 T^{6} - 731881822 T^{7} + 10961775828 T^{8} - 103764840872 T^{9} + 1353185769425 T^{10} - 11374809129845 T^{11} + 134695985357378 T^{12} - 1027001886511736 T^{13} + 11440507882026129 T^{14} - 81420620955130623 T^{15} + 872041110559406614 T^{16} - 81420620955130623 p T^{17} + 11440507882026129 p^{2} T^{18} - 1027001886511736 p^{3} T^{19} + 134695985357378 p^{4} T^{20} - 11374809129845 p^{5} T^{21} + 1353185769425 p^{6} T^{22} - 103764840872 p^{7} T^{23} + 10961775828 p^{8} T^{24} - 731881822 p^{9} T^{25} + 67555519 p^{10} T^{26} - 3713937 p^{11} T^{27} + 294238 p^{12} T^{28} - 12074 p^{13} T^{29} + 799 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 56 T + 2220 T^{2} - 63209 T^{3} + 1497077 T^{4} - 29816776 T^{5} + 523750220 T^{6} - 8188556978 T^{7} + 116694081790 T^{8} - 1528622743781 T^{9} + 18678888346970 T^{10} - 214455360319749 T^{11} + 2333837789894623 T^{12} - 24184707860680170 T^{13} + 239494812428915022 T^{14} - 2270495102884056799 T^{15} + 20615177670016269722 T^{16} - 2270495102884056799 p T^{17} + 239494812428915022 p^{2} T^{18} - 24184707860680170 p^{3} T^{19} + 2333837789894623 p^{4} T^{20} - 214455360319749 p^{5} T^{21} + 18678888346970 p^{6} T^{22} - 1528622743781 p^{7} T^{23} + 116694081790 p^{8} T^{24} - 8188556978 p^{9} T^{25} + 523750220 p^{10} T^{26} - 29816776 p^{11} T^{27} + 1497077 p^{12} T^{28} - 63209 p^{13} T^{29} + 2220 p^{14} T^{30} - 56 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 5 T + 619 T^{2} + 1789 T^{3} + 189875 T^{4} + 196844 T^{5} + 39294993 T^{6} - 23525976 T^{7} + 6261812710 T^{8} - 12148150242 T^{9} + 826900900131 T^{10} - 2371817048838 T^{11} + 94316828196477 T^{12} - 314513109217841 T^{13} + 9454726181158001 T^{14} - 32241383639665305 T^{15} + 835772322462085234 T^{16} - 32241383639665305 p T^{17} + 9454726181158001 p^{2} T^{18} - 314513109217841 p^{3} T^{19} + 94316828196477 p^{4} T^{20} - 2371817048838 p^{5} T^{21} + 826900900131 p^{6} T^{22} - 12148150242 p^{7} T^{23} + 6261812710 p^{8} T^{24} - 23525976 p^{9} T^{25} + 39294993 p^{10} T^{26} + 196844 p^{11} T^{27} + 189875 p^{12} T^{28} + 1789 p^{13} T^{29} + 619 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 7 T + 578 T^{2} + 3482 T^{3} + 170640 T^{4} + 922484 T^{5} + 35410568 T^{6} + 179607028 T^{7} + 5872703627 T^{8} + 28816908881 T^{9} + 823168298384 T^{10} + 3930733809449 T^{11} + 99853059470828 T^{12} + 460024212013485 T^{13} + 10626189131305926 T^{14} + 46732143059635336 T^{15} + 1002391679183032880 T^{16} + 46732143059635336 p T^{17} + 10626189131305926 p^{2} T^{18} + 460024212013485 p^{3} T^{19} + 99853059470828 p^{4} T^{20} + 3930733809449 p^{5} T^{21} + 823168298384 p^{6} T^{22} + 28816908881 p^{7} T^{23} + 5872703627 p^{8} T^{24} + 179607028 p^{9} T^{25} + 35410568 p^{10} T^{26} + 922484 p^{11} T^{27} + 170640 p^{12} T^{28} + 3482 p^{13} T^{29} + 578 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 35 T + 1878 T^{2} - 49186 T^{3} + 1529353 T^{4} - 32288323 T^{5} + 742439004 T^{6} - 13162301860 T^{7} + 244609704726 T^{8} - 3731903967818 T^{9} + 58541323697864 T^{10} - 780371593286307 T^{11} + 10585401042878535 T^{12} - 124417086367441608 T^{13} + 1479896786950776582 T^{14} - 15407385163006573471 T^{15} + \)\(16\!\cdots\!54\)\( T^{16} - 15407385163006573471 p T^{17} + 1479896786950776582 p^{2} T^{18} - 124417086367441608 p^{3} T^{19} + 10585401042878535 p^{4} T^{20} - 780371593286307 p^{5} T^{21} + 58541323697864 p^{6} T^{22} - 3731903967818 p^{7} T^{23} + 244609704726 p^{8} T^{24} - 13162301860 p^{9} T^{25} + 742439004 p^{10} T^{26} - 32288323 p^{11} T^{27} + 1529353 p^{12} T^{28} - 49186 p^{13} T^{29} + 1878 p^{14} T^{30} - 35 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−1.80398068785862196682353896443, −1.68216329232494230747754909904, −1.68028870177908769991618044062, −1.61896843403923117030432393856, −1.58521434858750801961188665354, −1.57878090744203103526579059495, −1.41084550519628252840304639903, −1.38675479236062530084153454472, −1.37093880512324081461755669231, −1.32449471375283707383060256271, −1.30545438173395465019744679204, −1.07357639585980724859272584588, −1.04566614850452164214363716816, −0.911544475283816709608484356413, −0.890283358461757745656255732122, −0.812534618725327065564116415666, −0.76939182166492468415381176489, −0.72924300520451592490539134939, −0.67204369383267270694003129666, −0.64509624445262566821255078282, −0.54023088655624391242276305905, −0.52675487589962159543976983720, −0.50923528040443416918459262167, −0.46134227473056029607218521630, −0.20339993281210689979117329034, 0.20339993281210689979117329034, 0.46134227473056029607218521630, 0.50923528040443416918459262167, 0.52675487589962159543976983720, 0.54023088655624391242276305905, 0.64509624445262566821255078282, 0.67204369383267270694003129666, 0.72924300520451592490539134939, 0.76939182166492468415381176489, 0.812534618725327065564116415666, 0.890283358461757745656255732122, 0.911544475283816709608484356413, 1.04566614850452164214363716816, 1.07357639585980724859272584588, 1.30545438173395465019744679204, 1.32449471375283707383060256271, 1.37093880512324081461755669231, 1.38675479236062530084153454472, 1.41084550519628252840304639903, 1.57878090744203103526579059495, 1.58521434858750801961188665354, 1.61896843403923117030432393856, 1.68028870177908769991618044062, 1.68216329232494230747754909904, 1.80398068785862196682353896443
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.