Dirichlet series
| L(s) = 1 | + 2·3-s − 6·7-s − 4·9-s − 8·13-s + 20·19-s − 12·21-s − 6·23-s − 8·27-s + 24·29-s + 8·31-s − 16·39-s + 6·41-s − 38·43-s − 10·49-s − 30·53-s + 40·57-s − 24·59-s − 32·61-s + 24·63-s − 22·67-s − 12·69-s + 44·73-s − 81-s + 48·87-s − 30·89-s + 48·91-s + 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.26·7-s − 4/3·9-s − 2.21·13-s + 4.58·19-s − 2.61·21-s − 1.25·23-s − 1.53·27-s + 4.45·29-s + 1.43·31-s − 2.56·39-s + 0.937·41-s − 5.79·43-s − 1.42·49-s − 4.12·53-s + 5.29·57-s − 3.12·59-s − 4.09·61-s + 3.02·63-s − 2.68·67-s − 1.44·69-s + 5.14·73-s − 1/9·81-s + 5.14·87-s − 3.17·89-s + 5.03·91-s + 1.65·93-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 5^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.11061\times 10^{20}\) |
| Root analytic conductor: | \(3.22188\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 5^{40} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03423939591\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03423939591\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 13 | \( 1 + 8 T + 19 T^{2} - 16 p T^{3} - 1631 T^{4} - 3712 T^{5} + 1572 p T^{6} + 155280 T^{7} + 321702 T^{8} - 1287464 T^{9} - 9005662 T^{10} - 1287464 p T^{11} + 321702 p^{2} T^{12} + 155280 p^{3} T^{13} + 1572 p^{5} T^{14} - 3712 p^{5} T^{15} - 1631 p^{6} T^{16} - 16 p^{8} T^{17} + 19 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| good | 3 | \( 1 - 2 T + 8 T^{2} - 16 T^{3} + 49 T^{4} - 44 T^{5} + 124 T^{6} + 34 T^{7} + 32 T^{8} + 1046 T^{9} - 476 T^{10} + 3940 T^{11} + 731 T^{12} + 6164 T^{13} + 20296 T^{14} + 4322 T^{15} + 27113 p T^{16} + 23584 T^{17} + 180992 T^{18} + 253300 T^{19} + 422848 T^{20} + 253300 p T^{21} + 180992 p^{2} T^{22} + 23584 p^{3} T^{23} + 27113 p^{5} T^{24} + 4322 p^{5} T^{25} + 20296 p^{6} T^{26} + 6164 p^{7} T^{27} + 731 p^{8} T^{28} + 3940 p^{9} T^{29} - 476 p^{10} T^{30} + 1046 p^{11} T^{31} + 32 p^{12} T^{32} + 34 p^{13} T^{33} + 124 p^{14} T^{34} - 44 p^{15} T^{35} + 49 p^{16} T^{36} - 16 p^{17} T^{37} + 8 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) |
| 7 | \( 1 + 6 T + 46 T^{2} + 204 T^{3} + 913 T^{4} + 3000 T^{5} + 9974 T^{6} + 3378 p T^{7} + 8312 p T^{8} + 68286 T^{9} + 18422 T^{10} - 760560 T^{11} - 2873485 T^{12} - 10928940 T^{13} - 22085254 T^{14} - 43630386 T^{15} + 24497555 T^{16} + 6837816 p^{2} T^{17} + 39044060 p^{2} T^{18} + 864239520 p T^{19} + 19009149104 T^{20} + 864239520 p^{2} T^{21} + 39044060 p^{4} T^{22} + 6837816 p^{5} T^{23} + 24497555 p^{4} T^{24} - 43630386 p^{5} T^{25} - 22085254 p^{6} T^{26} - 10928940 p^{7} T^{27} - 2873485 p^{8} T^{28} - 760560 p^{9} T^{29} + 18422 p^{10} T^{30} + 68286 p^{11} T^{31} + 8312 p^{13} T^{32} + 3378 p^{14} T^{33} + 9974 p^{14} T^{34} + 3000 p^{15} T^{35} + 913 p^{16} T^{36} + 204 p^{17} T^{37} + 46 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 12 T^{2} - 136 T^{3} + 173 T^{4} + 1368 T^{5} + 7748 T^{6} - 17696 T^{7} - 59988 T^{8} - 294592 T^{9} + 88124 p T^{10} + 808200 T^{11} + 6369555 T^{12} - 39806456 T^{13} + 114823164 T^{14} + 45121520 T^{15} + 804230751 T^{16} - 10245888432 T^{17} - 4864556040 T^{18} + 2591543776 T^{19} + 455569194888 T^{20} + 2591543776 p T^{21} - 4864556040 p^{2} T^{22} - 10245888432 p^{3} T^{23} + 804230751 p^{4} T^{24} + 45121520 p^{5} T^{25} + 114823164 p^{6} T^{26} - 39806456 p^{7} T^{27} + 6369555 p^{8} T^{28} + 808200 p^{9} T^{29} + 88124 p^{11} T^{30} - 294592 p^{11} T^{31} - 59988 p^{12} T^{32} - 17696 p^{13} T^{33} + 7748 p^{14} T^{34} + 1368 p^{15} T^{35} + 173 p^{16} T^{36} - 136 p^{17} T^{37} - 12 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 + 93 T^{2} + 200 T^{3} + 4742 T^{4} + 16620 T^{5} + 192887 T^{6} + 763996 T^{7} + 6582930 T^{8} + 26240888 T^{9} + 192638935 T^{10} + 744087576 T^{11} + 4916249424 T^{12} + 18446053528 T^{13} + 111859275351 T^{14} + 408656548664 T^{15} + 2319516531381 T^{16} + 8213633589876 T^{17} + 44342620885902 T^{18} + 151123179537052 T^{19} + 784591640152164 T^{20} + 151123179537052 p T^{21} + 44342620885902 p^{2} T^{22} + 8213633589876 p^{3} T^{23} + 2319516531381 p^{4} T^{24} + 408656548664 p^{5} T^{25} + 111859275351 p^{6} T^{26} + 18446053528 p^{7} T^{27} + 4916249424 p^{8} T^{28} + 744087576 p^{9} T^{29} + 192638935 p^{10} T^{30} + 26240888 p^{11} T^{31} + 6582930 p^{12} T^{32} + 763996 p^{13} T^{33} + 192887 p^{14} T^{34} + 16620 p^{15} T^{35} + 4742 p^{16} T^{36} + 200 p^{17} T^{37} + 93 p^{18} T^{38} + p^{20} T^{40} \) | |
| 19 | \( 1 - 20 T + 200 T^{2} - 1324 T^{3} + 6305 T^{4} - 19708 T^{5} + 9648 T^{6} + 364812 T^{7} - 2996684 T^{8} + 14899236 T^{9} - 49055496 T^{10} + 71491004 T^{11} + 305076391 T^{12} - 2724182156 T^{13} + 10376747448 T^{14} - 17634554644 T^{15} - 31523521753 T^{16} + 261749492872 T^{17} - 272912434152 T^{18} - 4535256149688 T^{19} + 31637774619416 T^{20} - 4535256149688 p T^{21} - 272912434152 p^{2} T^{22} + 261749492872 p^{3} T^{23} - 31523521753 p^{4} T^{24} - 17634554644 p^{5} T^{25} + 10376747448 p^{6} T^{26} - 2724182156 p^{7} T^{27} + 305076391 p^{8} T^{28} + 71491004 p^{9} T^{29} - 49055496 p^{10} T^{30} + 14899236 p^{11} T^{31} - 2996684 p^{12} T^{32} + 364812 p^{13} T^{33} + 9648 p^{14} T^{34} - 19708 p^{15} T^{35} + 6305 p^{16} T^{36} - 1324 p^{17} T^{37} + 200 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 6 T + 60 T^{2} + 472 T^{3} + 67 p T^{4} + 15564 T^{5} + 56288 T^{6} + 383546 T^{7} + 3212064 T^{8} + 12288910 T^{9} + 116001856 T^{10} + 428171196 T^{11} + 2516811927 T^{12} + 14740929380 T^{13} + 48968743092 T^{14} + 19838027654 p T^{15} + 67351182333 p T^{16} + 10136989611024 T^{17} + 51630488424000 T^{18} + 170044000980572 T^{19} + 1339699910780160 T^{20} + 170044000980572 p T^{21} + 51630488424000 p^{2} T^{22} + 10136989611024 p^{3} T^{23} + 67351182333 p^{5} T^{24} + 19838027654 p^{6} T^{25} + 48968743092 p^{6} T^{26} + 14740929380 p^{7} T^{27} + 2516811927 p^{8} T^{28} + 428171196 p^{9} T^{29} + 116001856 p^{10} T^{30} + 12288910 p^{11} T^{31} + 3212064 p^{12} T^{32} + 383546 p^{13} T^{33} + 56288 p^{14} T^{34} + 15564 p^{15} T^{35} + 67 p^{17} T^{36} + 472 p^{17} T^{37} + 60 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 24 T + 421 T^{2} - 5496 T^{3} + 60166 T^{4} - 580272 T^{5} + 5045467 T^{6} - 40862712 T^{7} + 310848154 T^{8} - 2248004928 T^{9} + 15606232171 T^{10} - 104285390304 T^{11} + 23365688796 p T^{12} - 4286978507184 T^{13} + 26500164246387 T^{14} - 160476015634656 T^{15} + 950174434733805 T^{16} - 5517893304075504 T^{17} + 31370889161022770 T^{18} - 174540194322426504 T^{19} + 951087271186793996 T^{20} - 174540194322426504 p T^{21} + 31370889161022770 p^{2} T^{22} - 5517893304075504 p^{3} T^{23} + 950174434733805 p^{4} T^{24} - 160476015634656 p^{5} T^{25} + 26500164246387 p^{6} T^{26} - 4286978507184 p^{7} T^{27} + 23365688796 p^{9} T^{28} - 104285390304 p^{9} T^{29} + 15606232171 p^{10} T^{30} - 2248004928 p^{11} T^{31} + 310848154 p^{12} T^{32} - 40862712 p^{13} T^{33} + 5045467 p^{14} T^{34} - 580272 p^{15} T^{35} + 60166 p^{16} T^{36} - 5496 p^{17} T^{37} + 421 p^{18} T^{38} - 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 8 T + 32 T^{2} - 176 T^{3} + 1474 T^{4} + 1008 T^{5} - 39744 T^{6} + 282024 T^{7} - 893475 T^{8} + 10418752 T^{9} - 45304352 T^{10} + 141870848 T^{11} + 339834808 T^{12} + 122939776 p T^{13} + 15206167200 T^{14} - 135003888000 T^{15} + 1299825695394 T^{16} - 1098261559056 T^{17} + 41305260704 p^{2} T^{18} - 1866052015648 p T^{19} + 596992660849420 T^{20} - 1866052015648 p^{2} T^{21} + 41305260704 p^{4} T^{22} - 1098261559056 p^{3} T^{23} + 1299825695394 p^{4} T^{24} - 135003888000 p^{5} T^{25} + 15206167200 p^{6} T^{26} + 122939776 p^{8} T^{27} + 339834808 p^{8} T^{28} + 141870848 p^{9} T^{29} - 45304352 p^{10} T^{30} + 10418752 p^{11} T^{31} - 893475 p^{12} T^{32} + 282024 p^{13} T^{33} - 39744 p^{14} T^{34} + 1008 p^{15} T^{35} + 1474 p^{16} T^{36} - 176 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + 85 T^{2} + 1714 T^{4} + 900 T^{5} + 18263 T^{6} - 158820 T^{7} + 3537818 T^{8} - 18459600 T^{9} + 125756399 T^{10} - 3477240 T^{11} + 176019644 T^{12} + 44706568608 T^{13} + 53858655431 T^{14} + 1062081220680 T^{15} + 4890160297469 T^{16} + 7788393745020 T^{17} - 130175969875618 T^{18} + 1819259131445508 T^{19} - 15056409000512812 T^{20} + 1819259131445508 p T^{21} - 130175969875618 p^{2} T^{22} + 7788393745020 p^{3} T^{23} + 4890160297469 p^{4} T^{24} + 1062081220680 p^{5} T^{25} + 53858655431 p^{6} T^{26} + 44706568608 p^{7} T^{27} + 176019644 p^{8} T^{28} - 3477240 p^{9} T^{29} + 125756399 p^{10} T^{30} - 18459600 p^{11} T^{31} + 3537818 p^{12} T^{32} - 158820 p^{13} T^{33} + 18263 p^{14} T^{34} + 900 p^{15} T^{35} + 1714 p^{16} T^{36} + 85 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( 1 - 6 T - 3 T^{2} + 14 T^{3} + 542 T^{4} - 8574 T^{5} + 39035 T^{6} + 13246 T^{7} - 2266662 T^{8} + 944678 T^{9} - 52593821 T^{10} + 1260672930 T^{11} - 2877013500 T^{12} + 3550963330 T^{13} + 70305369795 T^{14} + 53403765302 T^{15} - 4275020924019 T^{16} + 16170482313708 T^{17} + 304664606110338 T^{18} - 3265470233710004 T^{19} + 7397232150137580 T^{20} - 3265470233710004 p T^{21} + 304664606110338 p^{2} T^{22} + 16170482313708 p^{3} T^{23} - 4275020924019 p^{4} T^{24} + 53403765302 p^{5} T^{25} + 70305369795 p^{6} T^{26} + 3550963330 p^{7} T^{27} - 2877013500 p^{8} T^{28} + 1260672930 p^{9} T^{29} - 52593821 p^{10} T^{30} + 944678 p^{11} T^{31} - 2266662 p^{12} T^{32} + 13246 p^{13} T^{33} + 39035 p^{14} T^{34} - 8574 p^{15} T^{35} + 542 p^{16} T^{36} + 14 p^{17} T^{37} - 3 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 38 T + 692 T^{2} + 8388 T^{3} + 78225 T^{4} + 630944 T^{5} + 5124664 T^{6} + 45782858 T^{7} + 424729984 T^{8} + 3723134918 T^{9} + 29674959664 T^{10} + 220393752920 T^{11} + 1604436689227 T^{12} + 11880895193240 T^{13} + 2076226687588 p T^{14} + 660991005776570 T^{15} + 4724565729358987 T^{16} + 32552316081098408 T^{17} + 218616863784668584 T^{18} + 1448451960253708028 T^{19} + 9524575492737532480 T^{20} + 1448451960253708028 p T^{21} + 218616863784668584 p^{2} T^{22} + 32552316081098408 p^{3} T^{23} + 4724565729358987 p^{4} T^{24} + 660991005776570 p^{5} T^{25} + 2076226687588 p^{7} T^{26} + 11880895193240 p^{7} T^{27} + 1604436689227 p^{8} T^{28} + 220393752920 p^{9} T^{29} + 29674959664 p^{10} T^{30} + 3723134918 p^{11} T^{31} + 424729984 p^{12} T^{32} + 45782858 p^{13} T^{33} + 5124664 p^{14} T^{34} + 630944 p^{15} T^{35} + 78225 p^{16} T^{36} + 8388 p^{17} T^{37} + 692 p^{18} T^{38} + 38 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 716 T^{2} + 247582 T^{4} - 55074668 T^{6} + 8860554157 T^{8} - 1099150125680 T^{10} + 109466426764008 T^{12} - 8996850689415984 T^{14} + 622122718555939986 T^{16} - 36672815990399751592 T^{18} + \)\(18\!\cdots\!28\)\( T^{20} - 36672815990399751592 p^{2} T^{22} + 622122718555939986 p^{4} T^{24} - 8996850689415984 p^{6} T^{26} + 109466426764008 p^{8} T^{28} - 1099150125680 p^{10} T^{30} + 8860554157 p^{12} T^{32} - 55074668 p^{14} T^{34} + 247582 p^{16} T^{36} - 716 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 + 30 T + 450 T^{2} + 4688 T^{3} + 30623 T^{4} + 27840 T^{5} - 1956478 T^{6} - 29325938 T^{7} - 245842035 T^{8} - 983022592 T^{9} + 3958999240 T^{10} + 113257604496 T^{11} + 1111587083364 T^{12} + 5669475828592 T^{13} + 1238741370600 T^{14} - 302004862002064 T^{15} - 3366699031362654 T^{16} - 17986734578862612 T^{17} - 3779534396004 p^{2} T^{18} + 15846340427953136 p T^{19} + 9201530402326025274 T^{20} + 15846340427953136 p^{2} T^{21} - 3779534396004 p^{4} T^{22} - 17986734578862612 p^{3} T^{23} - 3366699031362654 p^{4} T^{24} - 302004862002064 p^{5} T^{25} + 1238741370600 p^{6} T^{26} + 5669475828592 p^{7} T^{27} + 1111587083364 p^{8} T^{28} + 113257604496 p^{9} T^{29} + 3958999240 p^{10} T^{30} - 983022592 p^{11} T^{31} - 245842035 p^{12} T^{32} - 29325938 p^{13} T^{33} - 1956478 p^{14} T^{34} + 27840 p^{15} T^{35} + 30623 p^{16} T^{36} + 4688 p^{17} T^{37} + 450 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 24 T + 312 T^{2} + 1920 T^{3} + 1637 T^{4} - 65040 T^{5} - 43944 T^{6} + 7327008 T^{7} + 75999012 T^{8} + 180424008 T^{9} - 2494237176 T^{10} - 17566470576 T^{11} + 73468519651 T^{12} + 1474747398408 T^{13} + 1314042455400 T^{14} - 98740841430528 T^{15} - 937792812440665 T^{16} - 3670565248494240 T^{17} - 11788702210662144 T^{18} - 189983577782759064 T^{19} - 2127066993729132552 T^{20} - 189983577782759064 p T^{21} - 11788702210662144 p^{2} T^{22} - 3670565248494240 p^{3} T^{23} - 937792812440665 p^{4} T^{24} - 98740841430528 p^{5} T^{25} + 1314042455400 p^{6} T^{26} + 1474747398408 p^{7} T^{27} + 73468519651 p^{8} T^{28} - 17566470576 p^{9} T^{29} - 2494237176 p^{10} T^{30} + 180424008 p^{11} T^{31} + 75999012 p^{12} T^{32} + 7327008 p^{13} T^{33} - 43944 p^{14} T^{34} - 65040 p^{15} T^{35} + 1637 p^{16} T^{36} + 1920 p^{17} T^{37} + 312 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 32 T + 309 T^{2} + 1184 T^{3} + 23830 T^{4} + 293856 T^{5} + 74331 T^{6} + 1144512 T^{7} + 144238458 T^{8} - 100479616 T^{9} - 1452703013 T^{10} + 97105941600 T^{11} + 295422401884 T^{12} + 3201395105792 T^{13} + 79720269437763 T^{14} + 218841954139296 T^{15} + 1685012516730189 T^{16} + 36417799395180288 T^{17} - 7027965470138142 T^{18} - 150118779762368736 T^{19} + 14121230121689336460 T^{20} - 150118779762368736 p T^{21} - 7027965470138142 p^{2} T^{22} + 36417799395180288 p^{3} T^{23} + 1685012516730189 p^{4} T^{24} + 218841954139296 p^{5} T^{25} + 79720269437763 p^{6} T^{26} + 3201395105792 p^{7} T^{27} + 295422401884 p^{8} T^{28} + 97105941600 p^{9} T^{29} - 1452703013 p^{10} T^{30} - 100479616 p^{11} T^{31} + 144238458 p^{12} T^{32} + 1144512 p^{13} T^{33} + 74331 p^{14} T^{34} + 293856 p^{15} T^{35} + 23830 p^{16} T^{36} + 1184 p^{17} T^{37} + 309 p^{18} T^{38} + 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 22 T - 186 T^{2} - 7372 T^{3} + 7529 T^{4} + 1289688 T^{5} + 1539806 T^{6} - 158249482 T^{7} - 250979712 T^{8} + 15623839014 T^{9} + 12116908710 T^{10} - 1322504583360 T^{11} + 1324093811859 T^{12} + 97746118979076 T^{13} - 346475613525966 T^{14} - 6113597414660818 T^{15} + 43515073927114955 T^{16} + 293792899866208584 T^{17} - 4024996710443013804 T^{18} - 7148218204523371416 T^{19} + \)\(29\!\cdots\!60\)\( T^{20} - 7148218204523371416 p T^{21} - 4024996710443013804 p^{2} T^{22} + 293792899866208584 p^{3} T^{23} + 43515073927114955 p^{4} T^{24} - 6113597414660818 p^{5} T^{25} - 346475613525966 p^{6} T^{26} + 97746118979076 p^{7} T^{27} + 1324093811859 p^{8} T^{28} - 1322504583360 p^{9} T^{29} + 12116908710 p^{10} T^{30} + 15623839014 p^{11} T^{31} - 250979712 p^{12} T^{32} - 158249482 p^{13} T^{33} + 1539806 p^{14} T^{34} + 1289688 p^{15} T^{35} + 7529 p^{16} T^{36} - 7372 p^{17} T^{37} - 186 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 24 T^{2} - 844 T^{3} + 5057 T^{4} - 81204 T^{5} + 239408 T^{6} - 1602608 T^{7} + 65836548 T^{8} - 344361208 T^{9} + 2796759688 T^{10} - 9051444444 T^{11} - 26670798969 T^{12} - 1240051540412 T^{13} + 2066999743704 T^{14} + 209381181811184 T^{15} - 968446193083017 T^{16} + 12112786141212408 T^{17} - 103979125688742648 T^{18} + 875405586564992248 T^{19} - 17318582476003527240 T^{20} + 875405586564992248 p T^{21} - 103979125688742648 p^{2} T^{22} + 12112786141212408 p^{3} T^{23} - 968446193083017 p^{4} T^{24} + 209381181811184 p^{5} T^{25} + 2066999743704 p^{6} T^{26} - 1240051540412 p^{7} T^{27} - 26670798969 p^{8} T^{28} - 9051444444 p^{9} T^{29} + 2796759688 p^{10} T^{30} - 344361208 p^{11} T^{31} + 65836548 p^{12} T^{32} - 1602608 p^{13} T^{33} + 239408 p^{14} T^{34} - 81204 p^{15} T^{35} + 5057 p^{16} T^{36} - 844 p^{17} T^{37} - 24 p^{18} T^{38} + p^{20} T^{40} \) | |
| 73 | \( ( 1 - 22 T + 607 T^{2} - 9394 T^{3} + 163369 T^{4} - 2025148 T^{5} + 27090660 T^{6} - 282198900 T^{7} + 3127501158 T^{8} - 27963745112 T^{9} + 264540654218 T^{10} - 27963745112 p T^{11} + 3127501158 p^{2} T^{12} - 282198900 p^{3} T^{13} + 27090660 p^{4} T^{14} - 2025148 p^{5} T^{15} + 163369 p^{6} T^{16} - 9394 p^{7} T^{17} + 607 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( 1 - 644 T^{2} + 209902 T^{4} - 47132548 T^{6} + 8304022349 T^{8} - 1225349555248 T^{10} + 156739679177192 T^{12} - 17731047061499440 T^{14} + 1796287612316104562 T^{16} - \)\(16\!\cdots\!16\)\( T^{18} + \)\(13\!\cdots\!32\)\( T^{20} - \)\(16\!\cdots\!16\)\( p^{2} T^{22} + 1796287612316104562 p^{4} T^{24} - 17731047061499440 p^{6} T^{26} + 156739679177192 p^{8} T^{28} - 1225349555248 p^{10} T^{30} + 8304022349 p^{12} T^{32} - 47132548 p^{14} T^{34} + 209902 p^{16} T^{36} - 644 p^{18} T^{38} + p^{20} T^{40} \) | |
| 83 | \( 1 - 764 T^{2} + 294598 T^{4} - 75345724 T^{6} + 14198423213 T^{8} - 2079116449456 T^{10} + 244129776508328 T^{12} - 23568594430322800 T^{14} + 1939524741825148514 T^{16} - \)\(14\!\cdots\!80\)\( T^{18} + \)\(11\!\cdots\!16\)\( T^{20} - \)\(14\!\cdots\!80\)\( p^{2} T^{22} + 1939524741825148514 p^{4} T^{24} - 23568594430322800 p^{6} T^{26} + 244129776508328 p^{8} T^{28} - 2079116449456 p^{10} T^{30} + 14198423213 p^{12} T^{32} - 75345724 p^{14} T^{34} + 294598 p^{16} T^{36} - 764 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( 1 + 30 T + 426 T^{2} + 2536 T^{3} - 12667 T^{4} - 434580 T^{5} - 4502746 T^{6} - 19169842 T^{7} + 188839572 T^{8} + 5060157454 T^{9} + 47659522294 T^{10} + 80516624172 T^{11} - 3655938268029 T^{12} - 45949579007044 T^{13} - 208631104420650 T^{14} + 1538783103054862 T^{15} + 39250049534247735 T^{16} + 391052871980802696 T^{17} + 1435712313070456356 T^{18} - 21100376261422886188 T^{19} - \)\(35\!\cdots\!44\)\( T^{20} - 21100376261422886188 p T^{21} + 1435712313070456356 p^{2} T^{22} + 391052871980802696 p^{3} T^{23} + 39250049534247735 p^{4} T^{24} + 1538783103054862 p^{5} T^{25} - 208631104420650 p^{6} T^{26} - 45949579007044 p^{7} T^{27} - 3655938268029 p^{8} T^{28} + 80516624172 p^{9} T^{29} + 47659522294 p^{10} T^{30} + 5060157454 p^{11} T^{31} + 188839572 p^{12} T^{32} - 19169842 p^{13} T^{33} - 4502746 p^{14} T^{34} - 434580 p^{15} T^{35} - 12667 p^{16} T^{36} + 2536 p^{17} T^{37} + 426 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 46 T + 366 T^{2} - 10596 T^{3} - 65583 T^{4} + 3816804 T^{5} + 21399646 T^{6} - 786984410 T^{7} - 1014859320 T^{8} + 171209682386 T^{9} - 144670956610 T^{10} - 26087650504956 T^{11} + 129598634965123 T^{12} + 3666774920683252 T^{13} - 27634411453738278 T^{14} - 364961841317653342 T^{15} + 4933074349426518371 T^{16} + 29980624341945527448 T^{17} - \)\(62\!\cdots\!04\)\( T^{18} - \)\(94\!\cdots\!16\)\( T^{19} + \)\(69\!\cdots\!24\)\( T^{20} - \)\(94\!\cdots\!16\)\( p T^{21} - \)\(62\!\cdots\!04\)\( p^{2} T^{22} + 29980624341945527448 p^{3} T^{23} + 4933074349426518371 p^{4} T^{24} - 364961841317653342 p^{5} T^{25} - 27634411453738278 p^{6} T^{26} + 3666774920683252 p^{7} T^{27} + 129598634965123 p^{8} T^{28} - 26087650504956 p^{9} T^{29} - 144670956610 p^{10} T^{30} + 171209682386 p^{11} T^{31} - 1014859320 p^{12} T^{32} - 786984410 p^{13} T^{33} + 21399646 p^{14} T^{34} + 3816804 p^{15} T^{35} - 65583 p^{16} T^{36} - 10596 p^{17} T^{37} + 366 p^{18} T^{38} + 46 p^{19} T^{39} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.12434069057748239107549972094, −1.93931164170902079681742283625, −1.86163527865705909867455311262, −1.80752743042546095409127455800, −1.80024148375230954953548143928, −1.72714482462767025645128508190, −1.66338501095126389636110894589, −1.60151427089190649719940325941, −1.59433330103280599851239759770, −1.50220375670041659734803941443, −1.44314454128430276559111683130, −1.42254070528595897190786670910, −1.39069946827864395313169225454, −1.15354888550821248725307146062, −1.13415582758398046439177721716, −1.10260208007439029240939361775, −0.946548241421064016477877057403, −0.76539498273064090100671429705, −0.70847116977019333008913140817, −0.70178025254273693930564547446, −0.43328996746068474684822022363, −0.41571491638685170732068968339, −0.16789914343852385680578954506, −0.06383695665394864105094917958, −0.06215021627278914990390745926, 0.06215021627278914990390745926, 0.06383695665394864105094917958, 0.16789914343852385680578954506, 0.41571491638685170732068968339, 0.43328996746068474684822022363, 0.70178025254273693930564547446, 0.70847116977019333008913140817, 0.76539498273064090100671429705, 0.946548241421064016477877057403, 1.10260208007439029240939361775, 1.13415582758398046439177721716, 1.15354888550821248725307146062, 1.39069946827864395313169225454, 1.42254070528595897190786670910, 1.44314454128430276559111683130, 1.50220375670041659734803941443, 1.59433330103280599851239759770, 1.60151427089190649719940325941, 1.66338501095126389636110894589, 1.72714482462767025645128508190, 1.80024148375230954953548143928, 1.80752743042546095409127455800, 1.86163527865705909867455311262, 1.93931164170902079681742283625, 2.12434069057748239107549972094
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.