Dirichlet series
| L(s) = 1 | + 2·2-s − 2·4-s + 3·5-s + 3·7-s − 6·8-s − 13·9-s + 6·10-s − 11-s + 22·13-s + 6·14-s − 16-s − 17-s − 26·18-s + 13·19-s − 6·20-s − 2·22-s − 2·23-s − 22·25-s + 44·26-s − 5·27-s − 6·28-s + 28·29-s + 12·31-s + 2·32-s − 2·34-s + 9·35-s + 26·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 4-s + 1.34·5-s + 1.13·7-s − 2.12·8-s − 4.33·9-s + 1.89·10-s − 0.301·11-s + 6.10·13-s + 1.60·14-s − 1/4·16-s − 0.242·17-s − 6.12·18-s + 2.98·19-s − 1.34·20-s − 0.426·22-s − 0.417·23-s − 4.39·25-s + 8.62·26-s − 0.962·27-s − 1.13·28-s + 5.19·29-s + 2.15·31-s + 0.353·32-s − 0.342·34-s + 1.52·35-s + 13/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(251^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(251^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(251^{17}\) |
| Sign: | $1$ |
| Analytic conductor: | \(135882.\) |
| Root analytic conductor: | \(1.41571\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((34,\ 251^{17} ,\ ( \ : [1/2]^{17} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.684844083\) |
| \(L(\frac12)\) | \(\approx\) | \(9.684844083\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 251 | \( ( 1 - T )^{17} \) |
| good | 2 | \( 1 - p T + 3 p T^{2} - 5 p T^{3} + 21 T^{4} - 15 p T^{5} + 55 T^{6} - 47 p T^{7} + 11 p^{4} T^{8} - 37 p^{3} T^{9} + 479 T^{10} - 93 p^{3} T^{11} + 529 p T^{12} - 195 p^{3} T^{13} + 575 p^{2} T^{14} - 57 p^{6} T^{15} + 677 p^{3} T^{16} - 125 p^{6} T^{17} + 677 p^{4} T^{18} - 57 p^{8} T^{19} + 575 p^{5} T^{20} - 195 p^{7} T^{21} + 529 p^{6} T^{22} - 93 p^{9} T^{23} + 479 p^{7} T^{24} - 37 p^{11} T^{25} + 11 p^{13} T^{26} - 47 p^{11} T^{27} + 55 p^{11} T^{28} - 15 p^{13} T^{29} + 21 p^{13} T^{30} - 5 p^{15} T^{31} + 3 p^{16} T^{32} - p^{17} T^{33} + p^{17} T^{34} \) |
| 3 | \( 1 + 13 T^{2} + 5 T^{3} + 32 p T^{4} + 68 T^{5} + 182 p T^{6} + 428 T^{7} + 2687 T^{8} + 1862 T^{9} + 3769 p T^{10} + 6854 T^{11} + 41095 T^{12} + 22552 T^{13} + 136637 T^{14} + 64016 T^{15} + 144613 p T^{16} + 180530 T^{17} + 144613 p^{2} T^{18} + 64016 p^{2} T^{19} + 136637 p^{3} T^{20} + 22552 p^{4} T^{21} + 41095 p^{5} T^{22} + 6854 p^{6} T^{23} + 3769 p^{8} T^{24} + 1862 p^{8} T^{25} + 2687 p^{9} T^{26} + 428 p^{10} T^{27} + 182 p^{12} T^{28} + 68 p^{12} T^{29} + 32 p^{14} T^{30} + 5 p^{14} T^{31} + 13 p^{15} T^{32} + p^{17} T^{34} \) | |
| 5 | \( 1 - 3 T + 31 T^{2} - 72 T^{3} + 468 T^{4} - 881 T^{5} + 4768 T^{6} - 7539 T^{7} + 39098 T^{8} - 55783 T^{9} + 286603 T^{10} - 389484 T^{11} + 1915432 T^{12} - 2487938 T^{13} + 11481759 T^{14} - 2813587 p T^{15} + 62122047 T^{16} - 72647307 T^{17} + 62122047 p T^{18} - 2813587 p^{3} T^{19} + 11481759 p^{3} T^{20} - 2487938 p^{4} T^{21} + 1915432 p^{5} T^{22} - 389484 p^{6} T^{23} + 286603 p^{7} T^{24} - 55783 p^{8} T^{25} + 39098 p^{9} T^{26} - 7539 p^{10} T^{27} + 4768 p^{11} T^{28} - 881 p^{12} T^{29} + 468 p^{13} T^{30} - 72 p^{14} T^{31} + 31 p^{15} T^{32} - 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 7 | \( 1 - 3 T + 48 T^{2} - 19 p T^{3} + 177 p T^{4} - 475 p T^{5} + 22870 T^{6} - 58943 T^{7} + 332802 T^{8} - 822312 T^{9} + 4028418 T^{10} - 9517541 T^{11} + 41695740 T^{12} - 93949998 T^{13} + 375908581 T^{14} - 804043343 T^{15} + 2979678299 T^{16} - 6011005895 T^{17} + 2979678299 p T^{18} - 804043343 p^{2} T^{19} + 375908581 p^{3} T^{20} - 93949998 p^{4} T^{21} + 41695740 p^{5} T^{22} - 9517541 p^{6} T^{23} + 4028418 p^{7} T^{24} - 822312 p^{8} T^{25} + 332802 p^{9} T^{26} - 58943 p^{10} T^{27} + 22870 p^{11} T^{28} - 475 p^{13} T^{29} + 177 p^{14} T^{30} - 19 p^{15} T^{31} + 48 p^{15} T^{32} - 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 11 | \( 1 + T + 65 T^{2} + 24 T^{3} + 2303 T^{4} + 274 T^{5} + 58221 T^{6} + 2576 T^{7} + 1134784 T^{8} + 81072 T^{9} + 18003396 T^{10} + 3578304 T^{11} + 22062955 p T^{12} + 8394442 p T^{13} + 2903678795 T^{14} + 1617752920 T^{15} + 32648883056 T^{16} + 20772298142 T^{17} + 32648883056 p T^{18} + 1617752920 p^{2} T^{19} + 2903678795 p^{3} T^{20} + 8394442 p^{5} T^{21} + 22062955 p^{6} T^{22} + 3578304 p^{6} T^{23} + 18003396 p^{7} T^{24} + 81072 p^{8} T^{25} + 1134784 p^{9} T^{26} + 2576 p^{10} T^{27} + 58221 p^{11} T^{28} + 274 p^{12} T^{29} + 2303 p^{13} T^{30} + 24 p^{14} T^{31} + 65 p^{15} T^{32} + p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 - 22 T + 327 T^{2} - 3591 T^{3} + 2498 p T^{4} - 248232 T^{5} + 1651854 T^{6} - 9644200 T^{7} + 3827079 p T^{8} - 225212378 T^{9} + 879770185 T^{10} - 2810241288 T^{11} + 472246003 p T^{12} + 1399315966 T^{13} - 109121956145 T^{14} + 58433464836 p T^{15} - 3699336995499 T^{16} + 14613844354636 T^{17} - 3699336995499 p T^{18} + 58433464836 p^{3} T^{19} - 109121956145 p^{3} T^{20} + 1399315966 p^{4} T^{21} + 472246003 p^{6} T^{22} - 2810241288 p^{6} T^{23} + 879770185 p^{7} T^{24} - 225212378 p^{8} T^{25} + 3827079 p^{10} T^{26} - 9644200 p^{10} T^{27} + 1651854 p^{11} T^{28} - 248232 p^{12} T^{29} + 2498 p^{14} T^{30} - 3591 p^{14} T^{31} + 327 p^{15} T^{32} - 22 p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 + T + 133 T^{2} + 268 T^{3} + 9244 T^{4} + 27575 T^{5} + 444762 T^{6} + 1692375 T^{7} + 16736710 T^{8} + 72663473 T^{9} + 524899057 T^{10} + 139501112 p T^{11} + 14112267644 T^{12} + 61982655932 T^{13} + 327056280789 T^{14} + 1344039683401 T^{15} + 6505937728089 T^{16} + 24716676967305 T^{17} + 6505937728089 p T^{18} + 1344039683401 p^{2} T^{19} + 327056280789 p^{3} T^{20} + 61982655932 p^{4} T^{21} + 14112267644 p^{5} T^{22} + 139501112 p^{7} T^{23} + 524899057 p^{7} T^{24} + 72663473 p^{8} T^{25} + 16736710 p^{9} T^{26} + 1692375 p^{10} T^{27} + 444762 p^{11} T^{28} + 27575 p^{12} T^{29} + 9244 p^{13} T^{30} + 268 p^{14} T^{31} + 133 p^{15} T^{32} + p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 - 13 T + 222 T^{2} - 2221 T^{3} + 23502 T^{4} - 194998 T^{5} + 1613988 T^{6} - 11594639 T^{7} + 81389501 T^{8} - 519469448 T^{9} + 3214276817 T^{10} - 18520540701 T^{11} + 103165666144 T^{12} - 542091355850 T^{13} + 2751364836018 T^{14} - 13263883720023 T^{15} + 61753665721869 T^{16} - 273906962104534 T^{17} + 61753665721869 p T^{18} - 13263883720023 p^{2} T^{19} + 2751364836018 p^{3} T^{20} - 542091355850 p^{4} T^{21} + 103165666144 p^{5} T^{22} - 18520540701 p^{6} T^{23} + 3214276817 p^{7} T^{24} - 519469448 p^{8} T^{25} + 81389501 p^{9} T^{26} - 11594639 p^{10} T^{27} + 1613988 p^{11} T^{28} - 194998 p^{12} T^{29} + 23502 p^{13} T^{30} - 2221 p^{14} T^{31} + 222 p^{15} T^{32} - 13 p^{16} T^{33} + p^{17} T^{34} \) | |
| 23 | \( 1 + 2 T + 246 T^{2} + 472 T^{3} + 30161 T^{4} + 55676 T^{5} + 2449686 T^{6} + 4351793 T^{7} + 147602921 T^{8} + 251800336 T^{9} + 7000359821 T^{10} + 11414484673 T^{11} + 270592769479 T^{12} + 418659247364 T^{13} + 8708044461887 T^{14} + 12658121552351 T^{15} + 236228585975581 T^{16} + 318600146073527 T^{17} + 236228585975581 p T^{18} + 12658121552351 p^{2} T^{19} + 8708044461887 p^{3} T^{20} + 418659247364 p^{4} T^{21} + 270592769479 p^{5} T^{22} + 11414484673 p^{6} T^{23} + 7000359821 p^{7} T^{24} + 251800336 p^{8} T^{25} + 147602921 p^{9} T^{26} + 4351793 p^{10} T^{27} + 2449686 p^{11} T^{28} + 55676 p^{12} T^{29} + 30161 p^{13} T^{30} + 472 p^{14} T^{31} + 246 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} \) | |
| 29 | \( 1 - 28 T + 602 T^{2} - 9400 T^{3} + 126452 T^{4} - 1451348 T^{5} + 15001278 T^{6} - 139469184 T^{7} + 1195787437 T^{8} - 9465739752 T^{9} + 70192831599 T^{10} - 488740210864 T^{11} + 3228475083182 T^{12} - 20297814483564 T^{13} + 122476315938600 T^{14} - 711585033189976 T^{15} + 4005438718327285 T^{16} - 21871939305117688 T^{17} + 4005438718327285 p T^{18} - 711585033189976 p^{2} T^{19} + 122476315938600 p^{3} T^{20} - 20297814483564 p^{4} T^{21} + 3228475083182 p^{5} T^{22} - 488740210864 p^{6} T^{23} + 70192831599 p^{7} T^{24} - 9465739752 p^{8} T^{25} + 1195787437 p^{9} T^{26} - 139469184 p^{10} T^{27} + 15001278 p^{11} T^{28} - 1451348 p^{12} T^{29} + 126452 p^{13} T^{30} - 9400 p^{14} T^{31} + 602 p^{15} T^{32} - 28 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 - 12 T + 361 T^{2} - 3663 T^{3} + 63568 T^{4} - 562300 T^{5} + 7276538 T^{6} - 57288091 T^{7} + 608911263 T^{8} - 4327577685 T^{9} + 39671337740 T^{10} - 257001286154 T^{11} + 2090299529775 T^{12} - 12421276415178 T^{13} + 91217472684149 T^{14} - 498870365298137 T^{15} + 3343830717932873 T^{16} - 16838912161949331 T^{17} + 3343830717932873 p T^{18} - 498870365298137 p^{2} T^{19} + 91217472684149 p^{3} T^{20} - 12421276415178 p^{4} T^{21} + 2090299529775 p^{5} T^{22} - 257001286154 p^{6} T^{23} + 39671337740 p^{7} T^{24} - 4327577685 p^{8} T^{25} + 608911263 p^{9} T^{26} - 57288091 p^{10} T^{27} + 7276538 p^{11} T^{28} - 562300 p^{12} T^{29} + 63568 p^{13} T^{30} - 3663 p^{14} T^{31} + 361 p^{15} T^{32} - 12 p^{16} T^{33} + p^{17} T^{34} \) | |
| 37 | \( 1 - 27 T + 17 p T^{2} - 10426 T^{3} + 153823 T^{4} - 1935194 T^{5} + 22315383 T^{6} - 232647766 T^{7} + 2267662294 T^{8} - 20538267480 T^{9} + 176186972446 T^{10} - 1425612728754 T^{11} + 11014966015165 T^{12} - 2188029697390 p T^{13} + 15427588621369 p T^{14} - 3845674010664990 T^{15} + 24911778549511730 T^{16} - 154483347658623386 T^{17} + 24911778549511730 p T^{18} - 3845674010664990 p^{2} T^{19} + 15427588621369 p^{4} T^{20} - 2188029697390 p^{5} T^{21} + 11014966015165 p^{5} T^{22} - 1425612728754 p^{6} T^{23} + 176186972446 p^{7} T^{24} - 20538267480 p^{8} T^{25} + 2267662294 p^{9} T^{26} - 232647766 p^{10} T^{27} + 22315383 p^{11} T^{28} - 1935194 p^{12} T^{29} + 153823 p^{13} T^{30} - 10426 p^{14} T^{31} + 17 p^{16} T^{32} - 27 p^{16} T^{33} + p^{17} T^{34} \) | |
| 41 | \( 1 + T + 370 T^{2} - 141 T^{3} + 66419 T^{4} - 126865 T^{5} + 7811356 T^{6} - 27158711 T^{7} + 693005062 T^{8} - 3345604560 T^{9} + 50737727474 T^{10} - 285769580321 T^{11} + 3222700257040 T^{12} - 18557932001698 T^{13} + 179352486137175 T^{14} - 974668139909303 T^{15} + 8626071655372911 T^{16} - 43172980681404095 T^{17} + 8626071655372911 p T^{18} - 974668139909303 p^{2} T^{19} + 179352486137175 p^{3} T^{20} - 18557932001698 p^{4} T^{21} + 3222700257040 p^{5} T^{22} - 285769580321 p^{6} T^{23} + 50737727474 p^{7} T^{24} - 3345604560 p^{8} T^{25} + 693005062 p^{9} T^{26} - 27158711 p^{10} T^{27} + 7811356 p^{11} T^{28} - 126865 p^{12} T^{29} + 66419 p^{13} T^{30} - 141 p^{14} T^{31} + 370 p^{15} T^{32} + p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 - 9 T + 381 T^{2} - 3330 T^{3} + 74587 T^{4} - 626370 T^{5} + 9994685 T^{6} - 79932102 T^{7} + 1022218640 T^{8} - 7745341120 T^{9} + 84285089900 T^{10} - 602966972786 T^{11} + 5783246060193 T^{12} - 38925473880766 T^{13} + 336561052878383 T^{14} - 2122402402822886 T^{15} + 16793877441281108 T^{16} - 98683122241581038 T^{17} + 16793877441281108 p T^{18} - 2122402402822886 p^{2} T^{19} + 336561052878383 p^{3} T^{20} - 38925473880766 p^{4} T^{21} + 5783246060193 p^{5} T^{22} - 602966972786 p^{6} T^{23} + 84285089900 p^{7} T^{24} - 7745341120 p^{8} T^{25} + 1022218640 p^{9} T^{26} - 79932102 p^{10} T^{27} + 9994685 p^{11} T^{28} - 626370 p^{12} T^{29} + 74587 p^{13} T^{30} - 3330 p^{14} T^{31} + 381 p^{15} T^{32} - 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 47 | \( 1 + 20 T + 539 T^{2} + 8097 T^{3} + 132115 T^{4} + 1605834 T^{5} + 20056685 T^{6} + 206148995 T^{7} + 2142835302 T^{8} + 19167731204 T^{9} + 173224474562 T^{10} + 1379601189521 T^{11} + 11194106900813 T^{12} + 81339011735318 T^{13} + 612242201931411 T^{14} + 4186001600199691 T^{15} + 30249408537814534 T^{16} + 200723203035174160 T^{17} + 30249408537814534 p T^{18} + 4186001600199691 p^{2} T^{19} + 612242201931411 p^{3} T^{20} + 81339011735318 p^{4} T^{21} + 11194106900813 p^{5} T^{22} + 1379601189521 p^{6} T^{23} + 173224474562 p^{7} T^{24} + 19167731204 p^{8} T^{25} + 2142835302 p^{9} T^{26} + 206148995 p^{10} T^{27} + 20056685 p^{11} T^{28} + 1605834 p^{12} T^{29} + 132115 p^{13} T^{30} + 8097 p^{14} T^{31} + 539 p^{15} T^{32} + 20 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 - T + 441 T^{2} + 322 T^{3} + 101481 T^{4} + 197494 T^{5} + 16305137 T^{6} + 842566 p T^{7} + 2032908942 T^{8} + 6502234464 T^{9} + 206464610670 T^{10} + 704884913034 T^{11} + 17527521175563 T^{12} + 60528664390346 T^{13} + 1262670466507339 T^{14} + 4244148057212694 T^{15} + 77865100767818918 T^{16} + 246569452473215298 T^{17} + 77865100767818918 p T^{18} + 4244148057212694 p^{2} T^{19} + 1262670466507339 p^{3} T^{20} + 60528664390346 p^{4} T^{21} + 17527521175563 p^{5} T^{22} + 704884913034 p^{6} T^{23} + 206464610670 p^{7} T^{24} + 6502234464 p^{8} T^{25} + 2032908942 p^{9} T^{26} + 842566 p^{11} T^{27} + 16305137 p^{11} T^{28} + 197494 p^{12} T^{29} + 101481 p^{13} T^{30} + 322 p^{14} T^{31} + 441 p^{15} T^{32} - p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 + 20 T + 734 T^{2} + 12408 T^{3} + 265020 T^{4} + 3848930 T^{5} + 62041502 T^{6} + 789606768 T^{7} + 10542318489 T^{8} + 119467599956 T^{9} + 1381468079645 T^{10} + 14098607773024 T^{11} + 144713205438578 T^{12} + 1340202733600798 T^{13} + 12387676412233544 T^{14} + 104573297813556984 T^{15} + 877737340473617637 T^{16} + 6765212874797431024 T^{17} + 877737340473617637 p T^{18} + 104573297813556984 p^{2} T^{19} + 12387676412233544 p^{3} T^{20} + 1340202733600798 p^{4} T^{21} + 144713205438578 p^{5} T^{22} + 14098607773024 p^{6} T^{23} + 1381468079645 p^{7} T^{24} + 119467599956 p^{8} T^{25} + 10542318489 p^{9} T^{26} + 789606768 p^{10} T^{27} + 62041502 p^{11} T^{28} + 3848930 p^{12} T^{29} + 265020 p^{13} T^{30} + 12408 p^{14} T^{31} + 734 p^{15} T^{32} + 20 p^{16} T^{33} + p^{17} T^{34} \) | |
| 61 | \( 1 - 59 T + 2112 T^{2} - 55415 T^{3} + 1177562 T^{4} - 21221476 T^{5} + 335021876 T^{6} - 4728841013 T^{7} + 60639557803 T^{8} - 714705093420 T^{9} + 7819314243897 T^{10} - 80060921611007 T^{11} + 773076350787456 T^{12} - 7089636137923804 T^{13} + 62176250216536602 T^{14} - 524679013289345893 T^{15} + 4282737142918878503 T^{16} - 33929936279905333490 T^{17} + 4282737142918878503 p T^{18} - 524679013289345893 p^{2} T^{19} + 62176250216536602 p^{3} T^{20} - 7089636137923804 p^{4} T^{21} + 773076350787456 p^{5} T^{22} - 80060921611007 p^{6} T^{23} + 7819314243897 p^{7} T^{24} - 714705093420 p^{8} T^{25} + 60639557803 p^{9} T^{26} - 4728841013 p^{10} T^{27} + 335021876 p^{11} T^{28} - 21221476 p^{12} T^{29} + 1177562 p^{13} T^{30} - 55415 p^{14} T^{31} + 2112 p^{15} T^{32} - 59 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 - 15 T + 739 T^{2} - 10078 T^{3} + 268868 T^{4} - 3345611 T^{5} + 64094448 T^{6} - 730468759 T^{7} + 11245254118 T^{8} - 117840395659 T^{9} + 1545847478289 T^{10} - 14948642880732 T^{11} + 172926454422394 T^{12} - 1547119860739074 T^{13} + 16118788783198125 T^{14} - 133524216140123901 T^{15} + 1269632332199075877 T^{16} - 9722118088107106935 T^{17} + 1269632332199075877 p T^{18} - 133524216140123901 p^{2} T^{19} + 16118788783198125 p^{3} T^{20} - 1547119860739074 p^{4} T^{21} + 172926454422394 p^{5} T^{22} - 14948642880732 p^{6} T^{23} + 1545847478289 p^{7} T^{24} - 117840395659 p^{8} T^{25} + 11245254118 p^{9} T^{26} - 730468759 p^{10} T^{27} + 64094448 p^{11} T^{28} - 3345611 p^{12} T^{29} + 268868 p^{13} T^{30} - 10078 p^{14} T^{31} + 739 p^{15} T^{32} - 15 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 + 26 T + 1007 T^{2} + 19849 T^{3} + 446241 T^{4} + 7194988 T^{5} + 120762359 T^{6} + 1665305267 T^{7} + 22992215838 T^{8} + 279200807408 T^{9} + 3339644446146 T^{10} + 36445396630849 T^{11} + 389968744645311 T^{12} + 3880328825560916 T^{13} + 37900927653943305 T^{14} + 347306265647546659 T^{15} + 3132356239603608570 T^{16} + 26587811659247253100 T^{17} + 3132356239603608570 p T^{18} + 347306265647546659 p^{2} T^{19} + 37900927653943305 p^{3} T^{20} + 3880328825560916 p^{4} T^{21} + 389968744645311 p^{5} T^{22} + 36445396630849 p^{6} T^{23} + 3339644446146 p^{7} T^{24} + 279200807408 p^{8} T^{25} + 22992215838 p^{9} T^{26} + 1665305267 p^{10} T^{27} + 120762359 p^{11} T^{28} + 7194988 p^{12} T^{29} + 446241 p^{13} T^{30} + 19849 p^{14} T^{31} + 1007 p^{15} T^{32} + 26 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 - 8 T + 581 T^{2} - 3373 T^{3} + 161686 T^{4} - 646934 T^{5} + 29717152 T^{6} - 74294097 T^{7} + 4174267013 T^{8} - 5314756935 T^{9} + 484566969456 T^{10} - 144409053834 T^{11} + 48456663741423 T^{12} + 21161696177878 T^{13} + 4284608404951523 T^{14} + 3849114453674043 T^{15} + 341995284726843055 T^{16} + 352083483795091715 T^{17} + 341995284726843055 p T^{18} + 3849114453674043 p^{2} T^{19} + 4284608404951523 p^{3} T^{20} + 21161696177878 p^{4} T^{21} + 48456663741423 p^{5} T^{22} - 144409053834 p^{6} T^{23} + 484566969456 p^{7} T^{24} - 5314756935 p^{8} T^{25} + 4174267013 p^{9} T^{26} - 74294097 p^{10} T^{27} + 29717152 p^{11} T^{28} - 646934 p^{12} T^{29} + 161686 p^{13} T^{30} - 3373 p^{14} T^{31} + 581 p^{15} T^{32} - 8 p^{16} T^{33} + p^{17} T^{34} \) | |
| 79 | \( 1 - 33 T + 1353 T^{2} - 32406 T^{3} + 807998 T^{4} - 15511805 T^{5} + 295820068 T^{6} - 4788030417 T^{7} + 75771882238 T^{8} - 1064743445583 T^{9} + 14544158476409 T^{10} - 180674985423206 T^{11} + 2177306441772962 T^{12} - 24183639832001318 T^{13} + 260411110563583929 T^{14} - 2603558555285058439 T^{15} + 25231939994033822625 T^{16} - \)\(22\!\cdots\!01\)\( T^{17} + 25231939994033822625 p T^{18} - 2603558555285058439 p^{2} T^{19} + 260411110563583929 p^{3} T^{20} - 24183639832001318 p^{4} T^{21} + 2177306441772962 p^{5} T^{22} - 180674985423206 p^{6} T^{23} + 14544158476409 p^{7} T^{24} - 1064743445583 p^{8} T^{25} + 75771882238 p^{9} T^{26} - 4788030417 p^{10} T^{27} + 295820068 p^{11} T^{28} - 15511805 p^{12} T^{29} + 807998 p^{13} T^{30} - 32406 p^{14} T^{31} + 1353 p^{15} T^{32} - 33 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 + 7 p T^{2} - 182 T^{3} + 180307 T^{4} - 64291 T^{5} + 39429864 T^{6} - 10639717 T^{7} + 6767265501 T^{8} - 978598957 T^{9} + 963090953338 T^{10} - 30597338308 T^{11} + 117355310194685 T^{12} + 5664290130289 T^{13} + 12476694291222476 T^{14} + 1130794417610287 T^{15} + 1169345388231792369 T^{16} + 116200485002181054 T^{17} + 1169345388231792369 p T^{18} + 1130794417610287 p^{2} T^{19} + 12476694291222476 p^{3} T^{20} + 5664290130289 p^{4} T^{21} + 117355310194685 p^{5} T^{22} - 30597338308 p^{6} T^{23} + 963090953338 p^{7} T^{24} - 978598957 p^{8} T^{25} + 6767265501 p^{9} T^{26} - 10639717 p^{10} T^{27} + 39429864 p^{11} T^{28} - 64291 p^{12} T^{29} + 180307 p^{13} T^{30} - 182 p^{14} T^{31} + 7 p^{16} T^{32} + p^{17} T^{34} \) | |
| 89 | \( 1 - 11 T + 1018 T^{2} - 10217 T^{3} + 499889 T^{4} - 4640361 T^{5} + 158874258 T^{6} - 1379474804 T^{7} + 36937331775 T^{8} - 302341492178 T^{9} + 6716122844629 T^{10} - 51998242411620 T^{11} + 994168111675971 T^{12} - 7269598825192290 T^{13} + 122782314292295471 T^{14} - 842720446308890714 T^{15} + 12831305239642471183 T^{16} - 81825398892517969620 T^{17} + 12831305239642471183 p T^{18} - 842720446308890714 p^{2} T^{19} + 122782314292295471 p^{3} T^{20} - 7269598825192290 p^{4} T^{21} + 994168111675971 p^{5} T^{22} - 51998242411620 p^{6} T^{23} + 6716122844629 p^{7} T^{24} - 302341492178 p^{8} T^{25} + 36937331775 p^{9} T^{26} - 1379474804 p^{10} T^{27} + 158874258 p^{11} T^{28} - 4640361 p^{12} T^{29} + 499889 p^{13} T^{30} - 10217 p^{14} T^{31} + 1018 p^{15} T^{32} - 11 p^{16} T^{33} + p^{17} T^{34} \) | |
| 97 | \( 1 + 10 T + 907 T^{2} + 8664 T^{3} + 415183 T^{4} + 3860446 T^{5} + 127714355 T^{6} + 1170394080 T^{7} + 29623108956 T^{8} + 268520719860 T^{9} + 5502372876200 T^{10} + 49124502273872 T^{11} + 847939461439269 T^{12} + 7377586202846146 T^{13} + 110804220390412409 T^{14} + 924869826743967992 T^{15} + 12437415583741272176 T^{16} + 97631790166212328132 T^{17} + 12437415583741272176 p T^{18} + 924869826743967992 p^{2} T^{19} + 110804220390412409 p^{3} T^{20} + 7377586202846146 p^{4} T^{21} + 847939461439269 p^{5} T^{22} + 49124502273872 p^{6} T^{23} + 5502372876200 p^{7} T^{24} + 268520719860 p^{8} T^{25} + 29623108956 p^{9} T^{26} + 1170394080 p^{10} T^{27} + 127714355 p^{11} T^{28} + 3860446 p^{12} T^{29} + 415183 p^{13} T^{30} + 8664 p^{14} T^{31} + 907 p^{15} T^{32} + 10 p^{16} T^{33} + p^{17} T^{34} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.43581921450324045129653015738, −3.26515364182586392006178598905, −3.21346567599327433812553061498, −3.18156806369292058979759049016, −3.04566942920996187009594413554, −2.96737079894138052927423536749, −2.89784822330670499483605025540, −2.73816874693695967756206202841, −2.62988865511031570602682339948, −2.53246658924143432405803228618, −2.39048770053119972087598500737, −2.31023932557365350115213195117, −2.16354721276676776172003352530, −2.14886766585876559915986079735, −2.13763892317394887526422250488, −1.97804318447035992397216255559, −1.60381253872383397080063001135, −1.56511750229428207299606285031, −1.52137311769411969876947501282, −1.23404376825392316087198645775, −1.17158717163194705061090649393, −1.03850445761253958232191636937, −0.842525288567058878804570448535, −0.75167137746730311708663471938, −0.39063239943440804145595494229, 0.39063239943440804145595494229, 0.75167137746730311708663471938, 0.842525288567058878804570448535, 1.03850445761253958232191636937, 1.17158717163194705061090649393, 1.23404376825392316087198645775, 1.52137311769411969876947501282, 1.56511750229428207299606285031, 1.60381253872383397080063001135, 1.97804318447035992397216255559, 2.13763892317394887526422250488, 2.14886766585876559915986079735, 2.16354721276676776172003352530, 2.31023932557365350115213195117, 2.39048770053119972087598500737, 2.53246658924143432405803228618, 2.62988865511031570602682339948, 2.73816874693695967756206202841, 2.89784822330670499483605025540, 2.96737079894138052927423536749, 3.04566942920996187009594413554, 3.18156806369292058979759049016, 3.21346567599327433812553061498, 3.26515364182586392006178598905, 3.43581921450324045129653015738
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.