Dirichlet series
| L(s) = 1 | + 4·2-s + 2·4-s + 5·5-s − 17·7-s − 12·8-s − 6·9-s + 20·10-s + 17·11-s + 13-s − 68·14-s − 19·16-s + 13·17-s − 24·18-s + 10·20-s + 68·22-s + 15·23-s − 9·25-s + 4·26-s − 2·27-s − 34·28-s + 25·29-s − 5·31-s + 52·34-s − 85·35-s − 12·36-s + 13·37-s − 60·40-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s + 2.23·5-s − 6.42·7-s − 4.24·8-s − 2·9-s + 6.32·10-s + 5.12·11-s + 0.277·13-s − 18.1·14-s − 4.75·16-s + 3.15·17-s − 5.65·18-s + 2.23·20-s + 14.4·22-s + 3.12·23-s − 9/5·25-s + 0.784·26-s − 0.384·27-s − 6.42·28-s + 4.64·29-s − 0.898·31-s + 8.91·34-s − 14.3·35-s − 2·36-s + 2.13·37-s − 9.48·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{17} \cdot 97^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{17} \cdot 97^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(7^{17} \cdot 97^{17}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.02340\times 10^{12}\) |
| Root analytic conductor: | \(2.32848\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((34,\ 7^{17} \cdot 97^{17} ,\ ( \ : [1/2]^{17} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(48.21739404\) |
| \(L(\frac12)\) | \(\approx\) | \(48.21739404\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + T )^{17} \) |
| 97 | \( ( 1 - T )^{17} \) | |
| good | 2 | \( 1 - p^{2} T + 7 p T^{2} - 9 p^{2} T^{3} + 87 T^{4} - 23 p^{3} T^{5} + 377 T^{6} - 719 T^{7} + 1337 T^{8} - 2355 T^{9} + 2029 p T^{10} - 1677 p^{2} T^{11} + 5453 p T^{12} - 17139 T^{13} + 13241 p T^{14} - 39637 T^{15} + 58263 T^{16} - 83137 T^{17} + 58263 p T^{18} - 39637 p^{2} T^{19} + 13241 p^{4} T^{20} - 17139 p^{4} T^{21} + 5453 p^{6} T^{22} - 1677 p^{8} T^{23} + 2029 p^{8} T^{24} - 2355 p^{8} T^{25} + 1337 p^{9} T^{26} - 719 p^{10} T^{27} + 377 p^{11} T^{28} - 23 p^{15} T^{29} + 87 p^{13} T^{30} - 9 p^{16} T^{31} + 7 p^{16} T^{32} - p^{18} T^{33} + p^{17} T^{34} \) |
| 3 | \( 1 + 2 p T^{2} + 2 T^{3} + 29 T^{4} + 4 p^{2} T^{5} + 91 T^{6} + 188 T^{7} + 332 T^{8} + 218 p T^{9} + 154 p^{2} T^{10} + 496 p T^{11} + 1733 p T^{12} + 4004 T^{13} + 4739 p T^{14} + 14722 T^{15} + 33931 T^{16} + 55700 T^{17} + 33931 p T^{18} + 14722 p^{2} T^{19} + 4739 p^{4} T^{20} + 4004 p^{4} T^{21} + 1733 p^{6} T^{22} + 496 p^{7} T^{23} + 154 p^{9} T^{24} + 218 p^{9} T^{25} + 332 p^{9} T^{26} + 188 p^{10} T^{27} + 91 p^{11} T^{28} + 4 p^{14} T^{29} + 29 p^{13} T^{30} + 2 p^{14} T^{31} + 2 p^{16} T^{32} + p^{17} T^{34} \) | |
| 5 | \( 1 - p T + 34 T^{2} - 128 T^{3} + 591 T^{4} - 1891 T^{5} + 6969 T^{6} - 20052 T^{7} + 64601 T^{8} - 171441 T^{9} + 500063 T^{10} - 1256098 T^{11} + 3409396 T^{12} - 8153322 T^{13} + 20835689 T^{14} - 47704972 T^{15} + 115269524 T^{16} - 2007982 p^{3} T^{17} + 115269524 p T^{18} - 47704972 p^{2} T^{19} + 20835689 p^{3} T^{20} - 8153322 p^{4} T^{21} + 3409396 p^{5} T^{22} - 1256098 p^{6} T^{23} + 500063 p^{7} T^{24} - 171441 p^{8} T^{25} + 64601 p^{9} T^{26} - 20052 p^{10} T^{27} + 6969 p^{11} T^{28} - 1891 p^{12} T^{29} + 591 p^{13} T^{30} - 128 p^{14} T^{31} + 34 p^{15} T^{32} - p^{17} T^{33} + p^{17} T^{34} \) | |
| 11 | \( 1 - 17 T + 211 T^{2} - 1866 T^{3} + 13769 T^{4} - 83998 T^{5} + 445517 T^{6} - 2028581 T^{7} + 8084856 T^{8} - 27408383 T^{9} + 77992681 T^{10} - 164207402 T^{11} + 151493685 T^{12} + 77397502 p T^{13} - 6289402681 T^{14} + 2624776683 p T^{15} - 107687518999 T^{16} + 373383972744 T^{17} - 107687518999 p T^{18} + 2624776683 p^{3} T^{19} - 6289402681 p^{3} T^{20} + 77397502 p^{5} T^{21} + 151493685 p^{5} T^{22} - 164207402 p^{6} T^{23} + 77992681 p^{7} T^{24} - 27408383 p^{8} T^{25} + 8084856 p^{9} T^{26} - 2028581 p^{10} T^{27} + 445517 p^{11} T^{28} - 83998 p^{12} T^{29} + 13769 p^{13} T^{30} - 1866 p^{14} T^{31} + 211 p^{15} T^{32} - 17 p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 - T + 79 T^{2} - 6 T^{3} + 3307 T^{4} + 2514 T^{5} + 98773 T^{6} + 141247 T^{7} + 2425790 T^{8} + 4398015 T^{9} + 51831823 T^{10} + 101184474 T^{11} + 971465505 T^{12} + 1927985450 T^{13} + 15947321717 T^{14} + 2433491169 p T^{15} + 231523301181 T^{16} + 445392455212 T^{17} + 231523301181 p T^{18} + 2433491169 p^{3} T^{19} + 15947321717 p^{3} T^{20} + 1927985450 p^{4} T^{21} + 971465505 p^{5} T^{22} + 101184474 p^{6} T^{23} + 51831823 p^{7} T^{24} + 4398015 p^{8} T^{25} + 2425790 p^{9} T^{26} + 141247 p^{10} T^{27} + 98773 p^{11} T^{28} + 2514 p^{12} T^{29} + 3307 p^{13} T^{30} - 6 p^{14} T^{31} + 79 p^{15} T^{32} - p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 - 13 T + 224 T^{2} - 2170 T^{3} + 22527 T^{4} - 178047 T^{5} + 1413576 T^{6} - 9558613 T^{7} + 3726191 p T^{8} - 377310605 T^{9} + 128390012 p T^{10} - 11684945617 T^{11} + 60634754803 T^{12} - 17420161616 p T^{13} + 1403637096144 T^{14} - 371881319219 p T^{15} + 27684464059729 T^{16} - 115714435078758 T^{17} + 27684464059729 p T^{18} - 371881319219 p^{3} T^{19} + 1403637096144 p^{3} T^{20} - 17420161616 p^{5} T^{21} + 60634754803 p^{5} T^{22} - 11684945617 p^{6} T^{23} + 128390012 p^{8} T^{24} - 377310605 p^{8} T^{25} + 3726191 p^{10} T^{26} - 9558613 p^{10} T^{27} + 1413576 p^{11} T^{28} - 178047 p^{12} T^{29} + 22527 p^{13} T^{30} - 2170 p^{14} T^{31} + 224 p^{15} T^{32} - 13 p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 + 154 T^{2} + 7 T^{3} + 11756 T^{4} + 7 p T^{5} + 594524 T^{6} - 41845 T^{7} + 22537845 T^{8} - 3920657 T^{9} + 690007852 T^{10} - 188992743 T^{11} + 17981881862 T^{12} - 6260513341 T^{13} + 414031542660 T^{14} - 159513836712 T^{15} + 8616491241819 T^{16} - 3319326869184 T^{17} + 8616491241819 p T^{18} - 159513836712 p^{2} T^{19} + 414031542660 p^{3} T^{20} - 6260513341 p^{4} T^{21} + 17981881862 p^{5} T^{22} - 188992743 p^{6} T^{23} + 690007852 p^{7} T^{24} - 3920657 p^{8} T^{25} + 22537845 p^{9} T^{26} - 41845 p^{10} T^{27} + 594524 p^{11} T^{28} + 7 p^{13} T^{29} + 11756 p^{13} T^{30} + 7 p^{14} T^{31} + 154 p^{15} T^{32} + p^{17} T^{34} \) | |
| 23 | \( 1 - 15 T + 360 T^{2} - 4038 T^{3} + 56312 T^{4} - 508804 T^{5} + 5270472 T^{6} - 39993900 T^{7} + 337485766 T^{8} - 2210215934 T^{9} + 15969465380 T^{10} - 4009245200 p T^{11} + 590552342688 T^{12} - 3069098439716 T^{13} + 17926124532968 T^{14} - 85673637438206 T^{15} + 466846282237053 T^{16} - 2090099919029142 T^{17} + 466846282237053 p T^{18} - 85673637438206 p^{2} T^{19} + 17926124532968 p^{3} T^{20} - 3069098439716 p^{4} T^{21} + 590552342688 p^{5} T^{22} - 4009245200 p^{7} T^{23} + 15969465380 p^{7} T^{24} - 2210215934 p^{8} T^{25} + 337485766 p^{9} T^{26} - 39993900 p^{10} T^{27} + 5270472 p^{11} T^{28} - 508804 p^{12} T^{29} + 56312 p^{13} T^{30} - 4038 p^{14} T^{31} + 360 p^{15} T^{32} - 15 p^{16} T^{33} + p^{17} T^{34} \) | |
| 29 | \( 1 - 25 T + 571 T^{2} - 8626 T^{3} + 119440 T^{4} - 1335087 T^{5} + 13891366 T^{6} - 124649022 T^{7} + 36370801 p T^{8} - 7935023096 T^{9} + 56984020665 T^{10} - 370331746154 T^{11} + 2332705421854 T^{12} - 13504064364601 T^{13} + 77567195804876 T^{14} - 14408093559166 p T^{15} + 2303594831101990 T^{16} - 12157726989936110 T^{17} + 2303594831101990 p T^{18} - 14408093559166 p^{3} T^{19} + 77567195804876 p^{3} T^{20} - 13504064364601 p^{4} T^{21} + 2332705421854 p^{5} T^{22} - 370331746154 p^{6} T^{23} + 56984020665 p^{7} T^{24} - 7935023096 p^{8} T^{25} + 36370801 p^{10} T^{26} - 124649022 p^{10} T^{27} + 13891366 p^{11} T^{28} - 1335087 p^{12} T^{29} + 119440 p^{13} T^{30} - 8626 p^{14} T^{31} + 571 p^{15} T^{32} - 25 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 + 5 T + 309 T^{2} + 1439 T^{3} + 47569 T^{4} + 209821 T^{5} + 4851495 T^{6} + 661262 p T^{7} + 368034801 T^{8} + 1499706690 T^{9} + 22113769399 T^{10} + 87052097632 T^{11} + 1094224458839 T^{12} + 4145191823555 T^{13} + 45730296092121 T^{14} + 165110744562063 T^{15} + 1639745758377042 T^{16} + 5558500865485234 T^{17} + 1639745758377042 p T^{18} + 165110744562063 p^{2} T^{19} + 45730296092121 p^{3} T^{20} + 4145191823555 p^{4} T^{21} + 1094224458839 p^{5} T^{22} + 87052097632 p^{6} T^{23} + 22113769399 p^{7} T^{24} + 1499706690 p^{8} T^{25} + 368034801 p^{9} T^{26} + 661262 p^{11} T^{27} + 4851495 p^{11} T^{28} + 209821 p^{12} T^{29} + 47569 p^{13} T^{30} + 1439 p^{14} T^{31} + 309 p^{15} T^{32} + 5 p^{16} T^{33} + p^{17} T^{34} \) | |
| 37 | \( 1 - 13 T + 378 T^{2} - 4390 T^{3} + 70324 T^{4} - 745870 T^{5} + 8733626 T^{6} - 84924200 T^{7} + 816899068 T^{8} - 7255029306 T^{9} + 61028406618 T^{10} - 493075332220 T^{11} + 3756064305510 T^{12} - 27595277275322 T^{13} + 193695832905932 T^{14} - 1298775899625062 T^{15} + 8444657333384951 T^{16} - 52003572984682914 T^{17} + 8444657333384951 p T^{18} - 1298775899625062 p^{2} T^{19} + 193695832905932 p^{3} T^{20} - 27595277275322 p^{4} T^{21} + 3756064305510 p^{5} T^{22} - 493075332220 p^{6} T^{23} + 61028406618 p^{7} T^{24} - 7255029306 p^{8} T^{25} + 816899068 p^{9} T^{26} - 84924200 p^{10} T^{27} + 8733626 p^{11} T^{28} - 745870 p^{12} T^{29} + 70324 p^{13} T^{30} - 4390 p^{14} T^{31} + 378 p^{15} T^{32} - 13 p^{16} T^{33} + p^{17} T^{34} \) | |
| 41 | \( 1 - 22 T + 625 T^{2} - 9590 T^{3} + 159932 T^{4} - 1918798 T^{5} + 23885127 T^{6} - 238430914 T^{7} + 2435113753 T^{8} - 21115833734 T^{9} + 187014402123 T^{10} - 1458104367730 T^{11} + 11638813485110 T^{12} - 83754473325932 T^{13} + 617745894641703 T^{14} - 4162589165002492 T^{15} + 28707311086295579 T^{16} - 181803666222552978 T^{17} + 28707311086295579 p T^{18} - 4162589165002492 p^{2} T^{19} + 617745894641703 p^{3} T^{20} - 83754473325932 p^{4} T^{21} + 11638813485110 p^{5} T^{22} - 1458104367730 p^{6} T^{23} + 187014402123 p^{7} T^{24} - 21115833734 p^{8} T^{25} + 2435113753 p^{9} T^{26} - 238430914 p^{10} T^{27} + 23885127 p^{11} T^{28} - 1918798 p^{12} T^{29} + 159932 p^{13} T^{30} - 9590 p^{14} T^{31} + 625 p^{15} T^{32} - 22 p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 - T + 265 T^{2} - 26 T^{3} + 35856 T^{4} + 30009 T^{5} + 3359694 T^{6} + 5694756 T^{7} + 248075893 T^{8} + 593964152 T^{9} + 15461514458 T^{10} + 44906859279 T^{11} + 847940220964 T^{12} + 2738076620510 T^{13} + 42119940016181 T^{14} + 3310998533647 p T^{15} + 1935220967794401 T^{16} + 6502180768661804 T^{17} + 1935220967794401 p T^{18} + 3310998533647 p^{3} T^{19} + 42119940016181 p^{3} T^{20} + 2738076620510 p^{4} T^{21} + 847940220964 p^{5} T^{22} + 44906859279 p^{6} T^{23} + 15461514458 p^{7} T^{24} + 593964152 p^{8} T^{25} + 248075893 p^{9} T^{26} + 5694756 p^{10} T^{27} + 3359694 p^{11} T^{28} + 30009 p^{12} T^{29} + 35856 p^{13} T^{30} - 26 p^{14} T^{31} + 265 p^{15} T^{32} - p^{16} T^{33} + p^{17} T^{34} \) | |
| 47 | \( 1 + 28 T + 807 T^{2} + 15390 T^{3} + 276390 T^{4} + 4119416 T^{5} + 57371555 T^{6} + 713768848 T^{7} + 8330518534 T^{8} + 89739472660 T^{9} + 911522720254 T^{10} + 8691272275636 T^{11} + 78470016629985 T^{12} + 671298015875048 T^{13} + 5454070018278956 T^{14} + 42201817353293406 T^{15} + 310663339275921510 T^{16} + 2182935050691288544 T^{17} + 310663339275921510 p T^{18} + 42201817353293406 p^{2} T^{19} + 5454070018278956 p^{3} T^{20} + 671298015875048 p^{4} T^{21} + 78470016629985 p^{5} T^{22} + 8691272275636 p^{6} T^{23} + 911522720254 p^{7} T^{24} + 89739472660 p^{8} T^{25} + 8330518534 p^{9} T^{26} + 713768848 p^{10} T^{27} + 57371555 p^{11} T^{28} + 4119416 p^{12} T^{29} + 276390 p^{13} T^{30} + 15390 p^{14} T^{31} + 807 p^{15} T^{32} + 28 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 - 36 T + 19 p T^{2} - 19750 T^{3} + 337841 T^{4} - 4872588 T^{5} + 64493632 T^{6} - 768590974 T^{7} + 8619620351 T^{8} - 89683775724 T^{9} + 889508578164 T^{10} - 8319292147546 T^{11} + 74781981001957 T^{12} - 640398682810292 T^{13} + 5299454715105059 T^{14} - 42034109691044176 T^{15} + 323168403819381749 T^{16} - 2388433040689870442 T^{17} + 323168403819381749 p T^{18} - 42034109691044176 p^{2} T^{19} + 5299454715105059 p^{3} T^{20} - 640398682810292 p^{4} T^{21} + 74781981001957 p^{5} T^{22} - 8319292147546 p^{6} T^{23} + 889508578164 p^{7} T^{24} - 89683775724 p^{8} T^{25} + 8619620351 p^{9} T^{26} - 768590974 p^{10} T^{27} + 64493632 p^{11} T^{28} - 4872588 p^{12} T^{29} + 337841 p^{13} T^{30} - 19750 p^{14} T^{31} + 19 p^{16} T^{32} - 36 p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 + 530 T^{2} + 584 T^{3} + 136447 T^{4} + 311330 T^{5} + 23229983 T^{6} + 79411917 T^{7} + 2998624641 T^{8} + 13067340491 T^{9} + 315963352279 T^{10} + 1577445951168 T^{11} + 28230452079959 T^{12} + 149854819736268 T^{13} + 2178578756546872 T^{14} + 11635231362230592 T^{15} + 146637180981427153 T^{16} + 751122860754152656 T^{17} + 146637180981427153 p T^{18} + 11635231362230592 p^{2} T^{19} + 2178578756546872 p^{3} T^{20} + 149854819736268 p^{4} T^{21} + 28230452079959 p^{5} T^{22} + 1577445951168 p^{6} T^{23} + 315963352279 p^{7} T^{24} + 13067340491 p^{8} T^{25} + 2998624641 p^{9} T^{26} + 79411917 p^{10} T^{27} + 23229983 p^{11} T^{28} + 311330 p^{12} T^{29} + 136447 p^{13} T^{30} + 584 p^{14} T^{31} + 530 p^{15} T^{32} + p^{17} T^{34} \) | |
| 61 | \( 1 - 22 T + 709 T^{2} - 12608 T^{3} + 238793 T^{4} - 3548730 T^{5} + 51303386 T^{6} - 654828903 T^{7} + 7956233270 T^{8} - 89304315280 T^{9} + 955554209558 T^{10} - 157828910143 p T^{11} + 93135132773588 T^{12} - 856844138234806 T^{13} + 7616997813223205 T^{14} - 64807981632923662 T^{15} + 534798778734660506 T^{16} - 4238781726532239124 T^{17} + 534798778734660506 p T^{18} - 64807981632923662 p^{2} T^{19} + 7616997813223205 p^{3} T^{20} - 856844138234806 p^{4} T^{21} + 93135132773588 p^{5} T^{22} - 157828910143 p^{7} T^{23} + 955554209558 p^{7} T^{24} - 89304315280 p^{8} T^{25} + 7956233270 p^{9} T^{26} - 654828903 p^{10} T^{27} + 51303386 p^{11} T^{28} - 3548730 p^{12} T^{29} + 238793 p^{13} T^{30} - 12608 p^{14} T^{31} + 709 p^{15} T^{32} - 22 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 + 15 T + 660 T^{2} + 8764 T^{3} + 219100 T^{4} + 2655782 T^{5} + 48868458 T^{6} + 547312420 T^{7} + 8185487660 T^{8} + 85189611332 T^{9} + 1090815140198 T^{10} + 10576559263604 T^{11} + 119631486570406 T^{12} + 16135213046014 p T^{13} + 11022575203598216 T^{14} + 92697093830016940 T^{15} + 863611993139568589 T^{16} + 6735843322832156954 T^{17} + 863611993139568589 p T^{18} + 92697093830016940 p^{2} T^{19} + 11022575203598216 p^{3} T^{20} + 16135213046014 p^{5} T^{21} + 119631486570406 p^{5} T^{22} + 10576559263604 p^{6} T^{23} + 1090815140198 p^{7} T^{24} + 85189611332 p^{8} T^{25} + 8185487660 p^{9} T^{26} + 547312420 p^{10} T^{27} + 48868458 p^{11} T^{28} + 2655782 p^{12} T^{29} + 219100 p^{13} T^{30} + 8764 p^{14} T^{31} + 660 p^{15} T^{32} + 15 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 - 16 T + 752 T^{2} - 9129 T^{3} + 253837 T^{4} - 2489155 T^{5} + 54021262 T^{6} - 446637660 T^{7} + 8455470029 T^{8} - 60748251946 T^{9} + 1058017118709 T^{10} - 6742416182310 T^{11} + 110919731570680 T^{12} - 637767607552317 T^{13} + 10040647875818103 T^{14} - 53052605697106341 T^{15} + 800881869955741959 T^{16} - 3961781041608903884 T^{17} + 800881869955741959 p T^{18} - 53052605697106341 p^{2} T^{19} + 10040647875818103 p^{3} T^{20} - 637767607552317 p^{4} T^{21} + 110919731570680 p^{5} T^{22} - 6742416182310 p^{6} T^{23} + 1058017118709 p^{7} T^{24} - 60748251946 p^{8} T^{25} + 8455470029 p^{9} T^{26} - 446637660 p^{10} T^{27} + 54021262 p^{11} T^{28} - 2489155 p^{12} T^{29} + 253837 p^{13} T^{30} - 9129 p^{14} T^{31} + 752 p^{15} T^{32} - 16 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 + 7 T + 915 T^{2} + 5718 T^{3} + 405965 T^{4} + 2288828 T^{5} + 116559251 T^{6} + 597806497 T^{7} + 24328117733 T^{8} + 114161916918 T^{9} + 3923675593737 T^{10} + 16903573026577 T^{11} + 506779069654133 T^{12} + 2006783991758636 T^{13} + 53564755312786595 T^{14} + 194791823191528168 T^{15} + 4691541828306436950 T^{16} + 15619744590074947894 T^{17} + 4691541828306436950 p T^{18} + 194791823191528168 p^{2} T^{19} + 53564755312786595 p^{3} T^{20} + 2006783991758636 p^{4} T^{21} + 506779069654133 p^{5} T^{22} + 16903573026577 p^{6} T^{23} + 3923675593737 p^{7} T^{24} + 114161916918 p^{8} T^{25} + 24328117733 p^{9} T^{26} + 597806497 p^{10} T^{27} + 116559251 p^{11} T^{28} + 2288828 p^{12} T^{29} + 405965 p^{13} T^{30} + 5718 p^{14} T^{31} + 915 p^{15} T^{32} + 7 p^{16} T^{33} + p^{17} T^{34} \) | |
| 79 | \( 1 - 27 T + 1258 T^{2} - 26378 T^{3} + 716775 T^{4} - 12468239 T^{5} + 253209867 T^{6} - 3790633174 T^{7} + 63032293221 T^{8} - 830437224111 T^{9} + 11828657091605 T^{10} - 139095241127356 T^{11} + 1741028560712020 T^{12} - 18433669639539974 T^{13} + 205882706645311247 T^{14} - 1972215076984154768 T^{15} + 19835340890329361924 T^{16} - \)\(17\!\cdots\!26\)\( T^{17} + 19835340890329361924 p T^{18} - 1972215076984154768 p^{2} T^{19} + 205882706645311247 p^{3} T^{20} - 18433669639539974 p^{4} T^{21} + 1741028560712020 p^{5} T^{22} - 139095241127356 p^{6} T^{23} + 11828657091605 p^{7} T^{24} - 830437224111 p^{8} T^{25} + 63032293221 p^{9} T^{26} - 3790633174 p^{10} T^{27} + 253209867 p^{11} T^{28} - 12468239 p^{12} T^{29} + 716775 p^{13} T^{30} - 26378 p^{14} T^{31} + 1258 p^{15} T^{32} - 27 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 + 29 T + 1357 T^{2} + 31106 T^{3} + 842136 T^{4} + 15997525 T^{5} + 323185440 T^{6} + 5249961284 T^{7} + 86877924255 T^{8} + 1233224318730 T^{9} + 17492969775520 T^{10} + 220249514071977 T^{11} + 2746136094473358 T^{12} + 30975685888835692 T^{13} + 344497991842182379 T^{14} + 3502225438554864255 T^{15} + 35035631810751496435 T^{16} + \)\(32\!\cdots\!40\)\( T^{17} + 35035631810751496435 p T^{18} + 3502225438554864255 p^{2} T^{19} + 344497991842182379 p^{3} T^{20} + 30975685888835692 p^{4} T^{21} + 2746136094473358 p^{5} T^{22} + 220249514071977 p^{6} T^{23} + 17492969775520 p^{7} T^{24} + 1233224318730 p^{8} T^{25} + 86877924255 p^{9} T^{26} + 5249961284 p^{10} T^{27} + 323185440 p^{11} T^{28} + 15997525 p^{12} T^{29} + 842136 p^{13} T^{30} + 31106 p^{14} T^{31} + 1357 p^{15} T^{32} + 29 p^{16} T^{33} + p^{17} T^{34} \) | |
| 89 | \( 1 - 73 T + 3285 T^{2} - 109082 T^{3} + 2956050 T^{4} - 68273891 T^{5} + 1387366584 T^{6} - 25287617184 T^{7} + 419874185049 T^{8} - 6420345532950 T^{9} + 91237179155133 T^{10} - 1213142990115308 T^{11} + 15178959445768612 T^{12} - 179468210968760109 T^{13} + 2011945889958474522 T^{14} - 21435316972393164922 T^{15} + \)\(21\!\cdots\!16\)\( T^{16} - \)\(21\!\cdots\!74\)\( T^{17} + \)\(21\!\cdots\!16\)\( p T^{18} - 21435316972393164922 p^{2} T^{19} + 2011945889958474522 p^{3} T^{20} - 179468210968760109 p^{4} T^{21} + 15178959445768612 p^{5} T^{22} - 1213142990115308 p^{6} T^{23} + 91237179155133 p^{7} T^{24} - 6420345532950 p^{8} T^{25} + 419874185049 p^{9} T^{26} - 25287617184 p^{10} T^{27} + 1387366584 p^{11} T^{28} - 68273891 p^{12} T^{29} + 2956050 p^{13} T^{30} - 109082 p^{14} T^{31} + 3285 p^{15} T^{32} - 73 p^{16} T^{33} + p^{17} T^{34} \) | |
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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
−2.90628096236201372928063110657, −2.69398503626409769580875705489, −2.64401970827152332808235435986, −2.54846679235345943669012266522, −2.34302583102378840922133509135, −2.30466296385263112769259970264, −2.27880663378262976993485461540, −2.19614136123905580398630734098, −2.16303939524049211037687535934, −2.10000214936213302492028640504, −2.07123087641941427790346742152, −1.85609138552988213747570830496, −1.63183253351251368627841743952, −1.57410536358172958154902108397, −1.38306892051588057304241881471, −1.26017272640857987822704067927, −1.15750485721412155541861454382, −1.12852810174433965259388780011, −1.03612693481573102923312716955, −0.820417273437329077280790539183, −0.78618526089941823900779895266, −0.67214464910765119875339091008, −0.58387173395501077492541278756, −0.46536243244622223312876046280, −0.26764847347901997752918229145, 0.26764847347901997752918229145, 0.46536243244622223312876046280, 0.58387173395501077492541278756, 0.67214464910765119875339091008, 0.78618526089941823900779895266, 0.820417273437329077280790539183, 1.03612693481573102923312716955, 1.12852810174433965259388780011, 1.15750485721412155541861454382, 1.26017272640857987822704067927, 1.38306892051588057304241881471, 1.57410536358172958154902108397, 1.63183253351251368627841743952, 1.85609138552988213747570830496, 2.07123087641941427790346742152, 2.10000214936213302492028640504, 2.16303939524049211037687535934, 2.19614136123905580398630734098, 2.27880663378262976993485461540, 2.30466296385263112769259970264, 2.34302583102378840922133509135, 2.54846679235345943669012266522, 2.64401970827152332808235435986, 2.69398503626409769580875705489, 2.90628096236201372928063110657
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.