Dirichlet series
| L(s) = 1 | − 6·4-s + 10·5-s + 24·7-s − 16·9-s − 2·11-s + 12·13-s + 10·16-s + 10·17-s + 18·19-s − 60·20-s + 12·23-s + 20·25-s + 4·27-s − 144·28-s + 17·29-s + 56·31-s − 8·32-s + 240·35-s + 96·36-s − 17·37-s + 5·41-s + 5·43-s + 12·44-s − 160·45-s + 30·47-s + 251·49-s − 72·52-s + ⋯ |
| L(s) = 1 | − 3·4-s + 4.47·5-s + 9.07·7-s − 5.33·9-s − 0.603·11-s + 3.32·13-s + 5/2·16-s + 2.42·17-s + 4.12·19-s − 13.4·20-s + 2.50·23-s + 4·25-s + 0.769·27-s − 27.2·28-s + 3.15·29-s + 10.0·31-s − 1.41·32-s + 40.5·35-s + 16·36-s − 2.79·37-s + 0.780·41-s + 0.762·43-s + 1.80·44-s − 23.8·45-s + 4.37·47-s + 35.8·49-s − 9.98·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{17} \cdot 37^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{17} \cdot 37^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(29^{17} \cdot 37^{17}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.22613\times 10^{15}\) |
| Root analytic conductor: | \(2.92710\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((34,\ 29^{17} \cdot 37^{17} ,\ ( \ : [1/2]^{17} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1780.004340\) |
| \(L(\frac12)\) | \(\approx\) | \(1780.004340\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 29 | \( ( 1 - T )^{17} \) |
| 37 | \( ( 1 + T )^{17} \) | |
| good | 2 | \( 1 + 3 p T^{2} + 13 p T^{4} + p^{3} T^{5} + 41 p T^{6} + 23 p T^{7} + 231 T^{8} + 3 p^{6} T^{9} + 583 T^{10} + 71 p^{3} T^{11} + 715 p T^{12} + 1483 T^{13} + 1657 p T^{14} + 3209 T^{15} + 7267 T^{16} + 1693 p^{2} T^{17} + 7267 p T^{18} + 3209 p^{2} T^{19} + 1657 p^{4} T^{20} + 1483 p^{4} T^{21} + 715 p^{6} T^{22} + 71 p^{9} T^{23} + 583 p^{7} T^{24} + 3 p^{14} T^{25} + 231 p^{9} T^{26} + 23 p^{11} T^{27} + 41 p^{12} T^{28} + p^{15} T^{29} + 13 p^{14} T^{30} + 3 p^{16} T^{32} + p^{17} T^{34} \) |
| 3 | \( 1 + 16 T^{2} - 4 T^{3} + 49 p T^{4} - 79 T^{5} + 983 T^{6} - 773 T^{7} + 1786 p T^{8} - 5204 T^{9} + 25195 T^{10} - 27101 T^{11} + 104705 T^{12} - 116554 T^{13} + 389038 T^{14} - 143111 p T^{15} + 1297165 T^{16} - 1376210 T^{17} + 1297165 p T^{18} - 143111 p^{3} T^{19} + 389038 p^{3} T^{20} - 116554 p^{4} T^{21} + 104705 p^{5} T^{22} - 27101 p^{6} T^{23} + 25195 p^{7} T^{24} - 5204 p^{8} T^{25} + 1786 p^{10} T^{26} - 773 p^{10} T^{27} + 983 p^{11} T^{28} - 79 p^{12} T^{29} + 49 p^{14} T^{30} - 4 p^{14} T^{31} + 16 p^{15} T^{32} + p^{17} T^{34} \) | |
| 5 | \( 1 - 2 p T + 16 p T^{2} - 452 T^{3} + 451 p T^{4} - 9509 T^{5} + 36976 T^{6} - 25886 p T^{7} + 427944 T^{8} - 52531 p^{2} T^{9} + 3862037 T^{10} - 10722782 T^{11} + 28796687 T^{12} - 73789548 T^{13} + 184100672 T^{14} - 88251032 p T^{15} + 1034198856 T^{16} - 2336771204 T^{17} + 1034198856 p T^{18} - 88251032 p^{3} T^{19} + 184100672 p^{3} T^{20} - 73789548 p^{4} T^{21} + 28796687 p^{5} T^{22} - 10722782 p^{6} T^{23} + 3862037 p^{7} T^{24} - 52531 p^{10} T^{25} + 427944 p^{9} T^{26} - 25886 p^{11} T^{27} + 36976 p^{11} T^{28} - 9509 p^{12} T^{29} + 451 p^{14} T^{30} - 452 p^{14} T^{31} + 16 p^{16} T^{32} - 2 p^{17} T^{33} + p^{17} T^{34} \) | |
| 7 | \( 1 - 24 T + 325 T^{2} - 3179 T^{3} + 24998 T^{4} - 166980 T^{5} + 980248 T^{6} - 5167501 T^{7} + 24830770 T^{8} - 15702262 p T^{9} + 451790505 T^{10} - 1734578722 T^{11} + 6248899053 T^{12} - 21195703159 T^{13} + 67866249498 T^{14} - 205510569036 T^{15} + 589318090414 T^{16} - 1601466652410 T^{17} + 589318090414 p T^{18} - 205510569036 p^{2} T^{19} + 67866249498 p^{3} T^{20} - 21195703159 p^{4} T^{21} + 6248899053 p^{5} T^{22} - 1734578722 p^{6} T^{23} + 451790505 p^{7} T^{24} - 15702262 p^{9} T^{25} + 24830770 p^{9} T^{26} - 5167501 p^{10} T^{27} + 980248 p^{11} T^{28} - 166980 p^{12} T^{29} + 24998 p^{13} T^{30} - 3179 p^{14} T^{31} + 325 p^{15} T^{32} - 24 p^{16} T^{33} + p^{17} T^{34} \) | |
| 11 | \( 1 + 2 T + 73 T^{2} + 201 T^{3} + 2742 T^{4} + 9558 T^{5} + 74446 T^{6} + 292917 T^{7} + 1675849 T^{8} + 6650733 T^{9} + 32484029 T^{10} + 121966077 T^{11} + 541423578 T^{12} + 1904617613 T^{13} + 7768267642 T^{14} + 25864809048 T^{15} + 97051680466 T^{16} + 305495562092 T^{17} + 97051680466 p T^{18} + 25864809048 p^{2} T^{19} + 7768267642 p^{3} T^{20} + 1904617613 p^{4} T^{21} + 541423578 p^{5} T^{22} + 121966077 p^{6} T^{23} + 32484029 p^{7} T^{24} + 6650733 p^{8} T^{25} + 1675849 p^{9} T^{26} + 292917 p^{10} T^{27} + 74446 p^{11} T^{28} + 9558 p^{12} T^{29} + 2742 p^{13} T^{30} + 201 p^{14} T^{31} + 73 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 - 12 T + 159 T^{2} - 1369 T^{3} + 11606 T^{4} - 81168 T^{5} + 545572 T^{6} - 3269818 T^{7} + 18773457 T^{8} - 7647649 p T^{9} + 505264815 T^{10} - 2408936913 T^{11} + 850050102 p T^{12} - 47994610895 T^{13} + 200996086242 T^{14} - 800884983828 T^{15} + 3081194619718 T^{16} - 11303452143696 T^{17} + 3081194619718 p T^{18} - 800884983828 p^{2} T^{19} + 200996086242 p^{3} T^{20} - 47994610895 p^{4} T^{21} + 850050102 p^{6} T^{22} - 2408936913 p^{6} T^{23} + 505264815 p^{7} T^{24} - 7647649 p^{9} T^{25} + 18773457 p^{9} T^{26} - 3269818 p^{10} T^{27} + 545572 p^{11} T^{28} - 81168 p^{12} T^{29} + 11606 p^{13} T^{30} - 1369 p^{14} T^{31} + 159 p^{15} T^{32} - 12 p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 - 10 T + 183 T^{2} - 1416 T^{3} + 15336 T^{4} - 100392 T^{5} + 831388 T^{6} - 4831736 T^{7} + 33645641 T^{8} - 178012740 T^{9} + 64162815 p T^{10} - 5319117078 T^{11} + 1729471280 p T^{12} - 132963037417 T^{13} + 672948417643 T^{14} - 2830701221638 T^{15} + 13237988354605 T^{16} - 51818082862906 T^{17} + 13237988354605 p T^{18} - 2830701221638 p^{2} T^{19} + 672948417643 p^{3} T^{20} - 132963037417 p^{4} T^{21} + 1729471280 p^{6} T^{22} - 5319117078 p^{6} T^{23} + 64162815 p^{8} T^{24} - 178012740 p^{8} T^{25} + 33645641 p^{9} T^{26} - 4831736 p^{10} T^{27} + 831388 p^{11} T^{28} - 100392 p^{12} T^{29} + 15336 p^{13} T^{30} - 1416 p^{14} T^{31} + 183 p^{15} T^{32} - 10 p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 - 18 T + 324 T^{2} - 3947 T^{3} + 44866 T^{4} - 22445 p T^{5} + 3782100 T^{6} - 29981642 T^{7} + 223405166 T^{8} - 1532136357 T^{9} + 9943453618 T^{10} - 60304273081 T^{11} + 347821759133 T^{12} - 1890802390654 T^{13} + 9808029664113 T^{14} - 48178395549842 T^{15} + 11908124998211 p T^{16} - 1008328661456760 T^{17} + 11908124998211 p^{2} T^{18} - 48178395549842 p^{2} T^{19} + 9808029664113 p^{3} T^{20} - 1890802390654 p^{4} T^{21} + 347821759133 p^{5} T^{22} - 60304273081 p^{6} T^{23} + 9943453618 p^{7} T^{24} - 1532136357 p^{8} T^{25} + 223405166 p^{9} T^{26} - 29981642 p^{10} T^{27} + 3782100 p^{11} T^{28} - 22445 p^{13} T^{29} + 44866 p^{13} T^{30} - 3947 p^{14} T^{31} + 324 p^{15} T^{32} - 18 p^{16} T^{33} + p^{17} T^{34} \) | |
| 23 | \( 1 - 12 T + 249 T^{2} - 2108 T^{3} + 25948 T^{4} - 168355 T^{5} + 1583396 T^{6} - 8079985 T^{7} + 65605119 T^{8} - 264585551 T^{9} + 2083953805 T^{10} - 6772021239 T^{11} + 58503371288 T^{12} - 166015364013 T^{13} + 1609261272872 T^{14} - 4348195014348 T^{15} + 42099541793038 T^{16} - 107716635869098 T^{17} + 42099541793038 p T^{18} - 4348195014348 p^{2} T^{19} + 1609261272872 p^{3} T^{20} - 166015364013 p^{4} T^{21} + 58503371288 p^{5} T^{22} - 6772021239 p^{6} T^{23} + 2083953805 p^{7} T^{24} - 264585551 p^{8} T^{25} + 65605119 p^{9} T^{26} - 8079985 p^{10} T^{27} + 1583396 p^{11} T^{28} - 168355 p^{12} T^{29} + 25948 p^{13} T^{30} - 2108 p^{14} T^{31} + 249 p^{15} T^{32} - 12 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 - 56 T + 1848 T^{2} - 44425 T^{3} + 857690 T^{4} - 13955098 T^{5} + 197394715 T^{6} - 2476882824 T^{7} + 27976868306 T^{8} - 287486908267 T^{9} + 2709474662440 T^{10} - 23565266476918 T^{11} + 190048702259414 T^{12} - 1426441132825191 T^{13} + 9992138518977189 T^{14} - 65458215947035537 T^{15} + 401591742175422781 T^{16} - 2309255519670786984 T^{17} + 401591742175422781 p T^{18} - 65458215947035537 p^{2} T^{19} + 9992138518977189 p^{3} T^{20} - 1426441132825191 p^{4} T^{21} + 190048702259414 p^{5} T^{22} - 23565266476918 p^{6} T^{23} + 2709474662440 p^{7} T^{24} - 287486908267 p^{8} T^{25} + 27976868306 p^{9} T^{26} - 2476882824 p^{10} T^{27} + 197394715 p^{11} T^{28} - 13955098 p^{12} T^{29} + 857690 p^{13} T^{30} - 44425 p^{14} T^{31} + 1848 p^{15} T^{32} - 56 p^{16} T^{33} + p^{17} T^{34} \) | |
| 41 | \( 1 - 5 T + 340 T^{2} - 1756 T^{3} + 58632 T^{4} - 318072 T^{5} + 6901357 T^{6} - 39034852 T^{7} + 625923082 T^{8} - 3617785162 T^{9} + 46511049401 T^{10} - 268308856994 T^{11} + 2923782035695 T^{12} - 16477113003810 T^{13} + 158147821143029 T^{14} - 854920349035846 T^{15} + 7430486712237455 T^{16} - 37872893770279294 T^{17} + 7430486712237455 p T^{18} - 854920349035846 p^{2} T^{19} + 158147821143029 p^{3} T^{20} - 16477113003810 p^{4} T^{21} + 2923782035695 p^{5} T^{22} - 268308856994 p^{6} T^{23} + 46511049401 p^{7} T^{24} - 3617785162 p^{8} T^{25} + 625923082 p^{9} T^{26} - 39034852 p^{10} T^{27} + 6901357 p^{11} T^{28} - 318072 p^{12} T^{29} + 58632 p^{13} T^{30} - 1756 p^{14} T^{31} + 340 p^{15} T^{32} - 5 p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 - 5 T + 565 T^{2} - 2608 T^{3} + 153667 T^{4} - 653789 T^{5} + 26799285 T^{6} - 104960821 T^{7} + 3367073215 T^{8} - 282048424 p T^{9} + 324416093855 T^{10} - 1073890675980 T^{11} + 24889369406324 T^{12} - 75656243219816 T^{13} + 1556530466239120 T^{14} - 4338746384619231 T^{15} + 80455375654848734 T^{16} - 205087556634743724 T^{17} + 80455375654848734 p T^{18} - 4338746384619231 p^{2} T^{19} + 1556530466239120 p^{3} T^{20} - 75656243219816 p^{4} T^{21} + 24889369406324 p^{5} T^{22} - 1073890675980 p^{6} T^{23} + 324416093855 p^{7} T^{24} - 282048424 p^{9} T^{25} + 3367073215 p^{9} T^{26} - 104960821 p^{10} T^{27} + 26799285 p^{11} T^{28} - 653789 p^{12} T^{29} + 153667 p^{13} T^{30} - 2608 p^{14} T^{31} + 565 p^{15} T^{32} - 5 p^{16} T^{33} + p^{17} T^{34} \) | |
| 47 | \( 1 - 30 T + 918 T^{2} - 17658 T^{3} + 326538 T^{4} - 4775739 T^{5} + 66985079 T^{6} - 805779366 T^{7} + 9350483298 T^{8} - 96833693697 T^{9} + 973795478391 T^{10} - 8939076477876 T^{11} + 80098898102426 T^{12} - 663812209436071 T^{13} + 5386793823247306 T^{14} - 40713731057053466 T^{15} + 301719002323904049 T^{16} - 2088605487372628514 T^{17} + 301719002323904049 p T^{18} - 40713731057053466 p^{2} T^{19} + 5386793823247306 p^{3} T^{20} - 663812209436071 p^{4} T^{21} + 80098898102426 p^{5} T^{22} - 8939076477876 p^{6} T^{23} + 973795478391 p^{7} T^{24} - 96833693697 p^{8} T^{25} + 9350483298 p^{9} T^{26} - 805779366 p^{10} T^{27} + 66985079 p^{11} T^{28} - 4775739 p^{12} T^{29} + 326538 p^{13} T^{30} - 17658 p^{14} T^{31} + 918 p^{15} T^{32} - 30 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 - 41 T + 1382 T^{2} - 33379 T^{3} + 702334 T^{4} - 12556733 T^{5} + 3820502 p T^{6} - 2928827659 T^{7} + 38996515526 T^{8} - 477436738423 T^{9} + 5448574966475 T^{10} - 57949842961274 T^{11} + 579059092049318 T^{12} - 5434763541748717 T^{13} + 48155262747403682 T^{14} - 402536151340593223 T^{15} + 3185056909485565420 T^{16} - 23823054174767211214 T^{17} + 3185056909485565420 p T^{18} - 402536151340593223 p^{2} T^{19} + 48155262747403682 p^{3} T^{20} - 5434763541748717 p^{4} T^{21} + 579059092049318 p^{5} T^{22} - 57949842961274 p^{6} T^{23} + 5448574966475 p^{7} T^{24} - 477436738423 p^{8} T^{25} + 38996515526 p^{9} T^{26} - 2928827659 p^{10} T^{27} + 3820502 p^{12} T^{28} - 12556733 p^{12} T^{29} + 702334 p^{13} T^{30} - 33379 p^{14} T^{31} + 1382 p^{15} T^{32} - 41 p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 - 24 T + 11 p T^{2} - 10933 T^{3} + 190218 T^{4} - 2605980 T^{5} + 36057501 T^{6} - 426413456 T^{7} + 5058258883 T^{8} - 53355845804 T^{9} + 563521565329 T^{10} - 5409188721667 T^{11} + 51988493860011 T^{12} - 460022352287178 T^{13} + 4078226101411569 T^{14} - 33532742124838208 T^{15} + 276395554878059885 T^{16} - 2120183649173700912 T^{17} + 276395554878059885 p T^{18} - 33532742124838208 p^{2} T^{19} + 4078226101411569 p^{3} T^{20} - 460022352287178 p^{4} T^{21} + 51988493860011 p^{5} T^{22} - 5409188721667 p^{6} T^{23} + 563521565329 p^{7} T^{24} - 53355845804 p^{8} T^{25} + 5058258883 p^{9} T^{26} - 426413456 p^{10} T^{27} + 36057501 p^{11} T^{28} - 2605980 p^{12} T^{29} + 190218 p^{13} T^{30} - 10933 p^{14} T^{31} + 11 p^{16} T^{32} - 24 p^{16} T^{33} + p^{17} T^{34} \) | |
| 61 | \( 1 - 28 T + 763 T^{2} - 13605 T^{3} + 227333 T^{4} - 3073894 T^{5} + 39504600 T^{6} - 443284884 T^{7} + 4822794003 T^{8} - 47745447936 T^{9} + 466856361903 T^{10} - 69793847194 p T^{11} + 38810448395130 T^{12} - 333939782078358 T^{13} + 2884557138957126 T^{14} - 23610656441011637 T^{15} + 194067571838339301 T^{16} - 1512712879810618864 T^{17} + 194067571838339301 p T^{18} - 23610656441011637 p^{2} T^{19} + 2884557138957126 p^{3} T^{20} - 333939782078358 p^{4} T^{21} + 38810448395130 p^{5} T^{22} - 69793847194 p^{7} T^{23} + 466856361903 p^{7} T^{24} - 47745447936 p^{8} T^{25} + 4822794003 p^{9} T^{26} - 443284884 p^{10} T^{27} + 39504600 p^{11} T^{28} - 3073894 p^{12} T^{29} + 227333 p^{13} T^{30} - 13605 p^{14} T^{31} + 763 p^{15} T^{32} - 28 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 - 6 T + 690 T^{2} - 3553 T^{3} + 222541 T^{4} - 968545 T^{5} + 673103 p T^{6} - 162932780 T^{7} + 6572120224 T^{8} - 19454947943 T^{9} + 754714671039 T^{10} - 1844694596909 T^{11} + 73191295446325 T^{12} - 153658619168314 T^{13} + 6245090697187345 T^{14} - 11846716378181834 T^{15} + 473388031773919408 T^{16} - 836394499065298520 T^{17} + 473388031773919408 p T^{18} - 11846716378181834 p^{2} T^{19} + 6245090697187345 p^{3} T^{20} - 153658619168314 p^{4} T^{21} + 73191295446325 p^{5} T^{22} - 1844694596909 p^{6} T^{23} + 754714671039 p^{7} T^{24} - 19454947943 p^{8} T^{25} + 6572120224 p^{9} T^{26} - 162932780 p^{10} T^{27} + 673103 p^{12} T^{28} - 968545 p^{12} T^{29} + 222541 p^{13} T^{30} - 3553 p^{14} T^{31} + 690 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 - 48 T + 1822 T^{2} - 48674 T^{3} + 1121735 T^{4} - 21622531 T^{5} + 374367253 T^{6} - 5747033996 T^{7} + 81126290737 T^{8} - 1045506249848 T^{9} + 12592339169048 T^{10} - 141081315758672 T^{11} + 1496943780487904 T^{12} - 14992283712647621 T^{13} + 143924011622104390 T^{14} - 1320066516208096386 T^{15} + 11719354280168680800 T^{16} - \)\(10\!\cdots\!56\)\( T^{17} + 11719354280168680800 p T^{18} - 1320066516208096386 p^{2} T^{19} + 143924011622104390 p^{3} T^{20} - 14992283712647621 p^{4} T^{21} + 1496943780487904 p^{5} T^{22} - 141081315758672 p^{6} T^{23} + 12592339169048 p^{7} T^{24} - 1045506249848 p^{8} T^{25} + 81126290737 p^{9} T^{26} - 5747033996 p^{10} T^{27} + 374367253 p^{11} T^{28} - 21622531 p^{12} T^{29} + 1121735 p^{13} T^{30} - 48674 p^{14} T^{31} + 1822 p^{15} T^{32} - 48 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 - 7 T + 9 p T^{2} - 5111 T^{3} + 224557 T^{4} - 1846347 T^{5} + 52616601 T^{6} - 440197667 T^{7} + 9399487876 T^{8} - 77743389633 T^{9} + 1350367936444 T^{10} - 10805194123183 T^{11} + 160729755734927 T^{12} - 1224347465190762 T^{13} + 16125990525547932 T^{14} - 115497137938817093 T^{15} + 1377049261700966005 T^{16} - 9172679560060000922 T^{17} + 1377049261700966005 p T^{18} - 115497137938817093 p^{2} T^{19} + 16125990525547932 p^{3} T^{20} - 1224347465190762 p^{4} T^{21} + 160729755734927 p^{5} T^{22} - 10805194123183 p^{6} T^{23} + 1350367936444 p^{7} T^{24} - 77743389633 p^{8} T^{25} + 9399487876 p^{9} T^{26} - 440197667 p^{10} T^{27} + 52616601 p^{11} T^{28} - 1846347 p^{12} T^{29} + 224557 p^{13} T^{30} - 5111 p^{14} T^{31} + 9 p^{16} T^{32} - 7 p^{16} T^{33} + p^{17} T^{34} \) | |
| 79 | \( 1 - 8 T + 308 T^{2} - 1597 T^{3} + 56416 T^{4} - 239574 T^{5} + 7147174 T^{6} - 20694335 T^{7} + 756050692 T^{8} - 2008971369 T^{9} + 70752615867 T^{10} - 147393392397 T^{11} + 6137092246352 T^{12} - 12658917419943 T^{13} + 484337232998102 T^{14} - 580002944481983 T^{15} + 36843700371570806 T^{16} - 47728687195455380 T^{17} + 36843700371570806 p T^{18} - 580002944481983 p^{2} T^{19} + 484337232998102 p^{3} T^{20} - 12658917419943 p^{4} T^{21} + 6137092246352 p^{5} T^{22} - 147393392397 p^{6} T^{23} + 70752615867 p^{7} T^{24} - 2008971369 p^{8} T^{25} + 756050692 p^{9} T^{26} - 20694335 p^{10} T^{27} + 7147174 p^{11} T^{28} - 239574 p^{12} T^{29} + 56416 p^{13} T^{30} - 1597 p^{14} T^{31} + 308 p^{15} T^{32} - 8 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 - 54 T + 2193 T^{2} - 64757 T^{3} + 1624636 T^{4} - 34707656 T^{5} + 662099223 T^{6} - 11337852275 T^{7} + 178113708501 T^{8} - 2578658963856 T^{9} + 34835261338748 T^{10} - 440607260977460 T^{11} + 5259659393894011 T^{12} - 59387879000593319 T^{13} + 637626606627773593 T^{14} - 6516003115779939384 T^{15} + 63589383491287033776 T^{16} - \)\(59\!\cdots\!70\)\( T^{17} + 63589383491287033776 p T^{18} - 6516003115779939384 p^{2} T^{19} + 637626606627773593 p^{3} T^{20} - 59387879000593319 p^{4} T^{21} + 5259659393894011 p^{5} T^{22} - 440607260977460 p^{6} T^{23} + 34835261338748 p^{7} T^{24} - 2578658963856 p^{8} T^{25} + 178113708501 p^{9} T^{26} - 11337852275 p^{10} T^{27} + 662099223 p^{11} T^{28} - 34707656 p^{12} T^{29} + 1624636 p^{13} T^{30} - 64757 p^{14} T^{31} + 2193 p^{15} T^{32} - 54 p^{16} T^{33} + p^{17} T^{34} \) | |
| 89 | \( 1 - 26 T + 1200 T^{2} - 22460 T^{3} + 599085 T^{4} - 8693267 T^{5} + 172843149 T^{6} - 2012057856 T^{7} + 33263452820 T^{8} - 315197979029 T^{9} + 4692860706199 T^{10} - 36428056744298 T^{11} + 529400758461234 T^{12} - 3410018267946946 T^{13} + 52167143052066175 T^{14} - 290525305404796536 T^{15} + 4812277869767324673 T^{16} - 25113129576802735580 T^{17} + 4812277869767324673 p T^{18} - 290525305404796536 p^{2} T^{19} + 52167143052066175 p^{3} T^{20} - 3410018267946946 p^{4} T^{21} + 529400758461234 p^{5} T^{22} - 36428056744298 p^{6} T^{23} + 4692860706199 p^{7} T^{24} - 315197979029 p^{8} T^{25} + 33263452820 p^{9} T^{26} - 2012057856 p^{10} T^{27} + 172843149 p^{11} T^{28} - 8693267 p^{12} T^{29} + 599085 p^{13} T^{30} - 22460 p^{14} T^{31} + 1200 p^{15} T^{32} - 26 p^{16} T^{33} + p^{17} T^{34} \) | |
| 97 | \( 1 - 48 T + 2199 T^{2} - 67348 T^{3} + 1902585 T^{4} - 44301924 T^{5} + 957658953 T^{6} - 18277649883 T^{7} + 326666029430 T^{8} - 5318310636056 T^{9} + 81638885327726 T^{10} - 1160641913871775 T^{11} + 15634012734815867 T^{12} - 196876857924227522 T^{13} + 2356577021910339018 T^{14} - 26508332309691540698 T^{15} + \)\(28\!\cdots\!13\)\( T^{16} - \)\(28\!\cdots\!04\)\( T^{17} + \)\(28\!\cdots\!13\)\( p T^{18} - 26508332309691540698 p^{2} T^{19} + 2356577021910339018 p^{3} T^{20} - 196876857924227522 p^{4} T^{21} + 15634012734815867 p^{5} T^{22} - 1160641913871775 p^{6} T^{23} + 81638885327726 p^{7} T^{24} - 5318310636056 p^{8} T^{25} + 326666029430 p^{9} T^{26} - 18277649883 p^{10} T^{27} + 957658953 p^{11} T^{28} - 44301924 p^{12} T^{29} + 1902585 p^{13} T^{30} - 67348 p^{14} T^{31} + 2199 p^{15} T^{32} - 48 p^{16} T^{33} + p^{17} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.37430138381134405516879198161, −2.30091386092347329643521981054, −2.23926724314646346746612401606, −2.18280171510462633344964807152, −2.14238302132909652839272898238, −2.10861952673092755416131912897, −2.03364396455953434075842178754, −1.92521876316598303548307967377, −1.84079614152725579456472871175, −1.73612242637596407623939472201, −1.66275889124273940810017403524, −1.65969162374842776134320854960, −1.52277202526850512794213574698, −1.24865017605966661745415146754, −1.23264370065561655814472874494, −1.17256053217238529545096953889, −1.00466863696994001013258821595, −0.961696055761155118765910071087, −0.958050068806704057039719343278, −0.869167031666698927825498358011, −0.75595163485241039684300379777, −0.72878639066981993489190894274, −0.67237227761334590898254375779, −0.61704784548731698487402135821, −0.61102112900172808563829462996, 0.61102112900172808563829462996, 0.61704784548731698487402135821, 0.67237227761334590898254375779, 0.72878639066981993489190894274, 0.75595163485241039684300379777, 0.869167031666698927825498358011, 0.958050068806704057039719343278, 0.961696055761155118765910071087, 1.00466863696994001013258821595, 1.17256053217238529545096953889, 1.23264370065561655814472874494, 1.24865017605966661745415146754, 1.52277202526850512794213574698, 1.65969162374842776134320854960, 1.66275889124273940810017403524, 1.73612242637596407623939472201, 1.84079614152725579456472871175, 1.92521876316598303548307967377, 2.03364396455953434075842178754, 2.10861952673092755416131912897, 2.14238302132909652839272898238, 2.18280171510462633344964807152, 2.23926724314646346746612401606, 2.30091386092347329643521981054, 2.37430138381134405516879198161
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.