# Properties

 Label 40-65e20-1.1-c1e20-0-1 Degree $40$ Conductor $1.812\times 10^{36}$ Sign $1$ Analytic cond. $2.01284\times 10^{-6}$ Root an. cond. $0.720435$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s − 2·3-s + 15·4-s − 6·5-s + 8·6-s − 6·7-s − 40·8-s − 4·9-s + 24·10-s − 16·11-s − 30·12-s + 2·13-s + 24·14-s + 12·15-s + 99·16-s − 10·17-s + 16·18-s + 20·19-s − 90·20-s + 12·21-s + 64·22-s − 2·23-s + 80·24-s + 9·25-s − 8·26-s + 16·27-s − 90·28-s + ⋯
 L(s)  = 1 − 2.82·2-s − 1.15·3-s + 15/2·4-s − 2.68·5-s + 3.26·6-s − 2.26·7-s − 14.1·8-s − 4/3·9-s + 7.58·10-s − 4.82·11-s − 8.66·12-s + 0.554·13-s + 6.41·14-s + 3.09·15-s + 99/4·16-s − 2.42·17-s + 3.77·18-s + 4.58·19-s − 20.1·20-s + 2.61·21-s + 13.6·22-s − 0.417·23-s + 16.3·24-s + 9/5·25-s − 1.56·26-s + 3.07·27-s − 17.0·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \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(5^{20} \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: $$5^{20} \cdot 13^{20}$$ Sign: $1$ Analytic conductor: $$2.01284\times 10^{-6}$$ Root analytic conductor: $$0.720435$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{65} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(40,\ 5^{20} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.01121198731$$ $$L(\frac12)$$ $$\approx$$ $$0.01121198731$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + 6 T + 27 T^{2} + 114 T^{3} + 393 T^{4} + 1244 T^{5} + 3672 T^{6} + 10116 T^{7} + 25998 T^{8} + 63264 T^{9} + 146954 T^{10} + 63264 p T^{11} + 25998 p^{2} T^{12} + 10116 p^{3} T^{13} + 3672 p^{4} T^{14} + 1244 p^{5} T^{15} + 393 p^{6} T^{16} + 114 p^{7} T^{17} + 27 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20}$$
13 $$1 - 2 T + 15 T^{2} - 2 p T^{3} + 145 T^{4} - 944 T^{5} + 1124 T^{6} + 16 p T^{7} - 9610 T^{8} + 74660 T^{9} + 342346 T^{10} + 74660 p T^{11} - 9610 p^{2} T^{12} + 16 p^{4} T^{13} + 1124 p^{4} T^{14} - 944 p^{5} T^{15} + 145 p^{6} T^{16} - 2 p^{8} T^{17} + 15 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20}$$
good2 $$1 + p^{2} T + T^{2} - p^{4} T^{3} - 9 p T^{4} + 11 p T^{5} + 45 T^{6} + 5 p T^{7} - 15 T^{8} - 11 p^{2} T^{9} - 9 p^{4} T^{10} - 45 p^{2} T^{11} + 41 T^{12} + 151 p^{2} T^{13} + 1081 T^{14} - p^{5} T^{15} - 1375 p T^{16} - 1397 p T^{17} + 1609 T^{18} + 1915 p T^{19} + 2909 T^{20} + 1915 p^{2} T^{21} + 1609 p^{2} T^{22} - 1397 p^{4} T^{23} - 1375 p^{5} T^{24} - p^{10} T^{25} + 1081 p^{6} T^{26} + 151 p^{9} T^{27} + 41 p^{8} T^{28} - 45 p^{11} T^{29} - 9 p^{14} T^{30} - 11 p^{13} T^{31} - 15 p^{12} T^{32} + 5 p^{14} T^{33} + 45 p^{14} T^{34} + 11 p^{16} T^{35} - 9 p^{17} T^{36} - p^{21} T^{37} + p^{18} T^{38} + p^{21} T^{39} + p^{20} T^{40}$$
3 $$1 + 2 T + 8 T^{2} + 8 T^{3} + 25 T^{4} + 20 T^{5} + 76 T^{6} + 14 p^{2} T^{7} + 352 T^{8} + 730 T^{9} + 1348 T^{10} + 2660 T^{11} + 4147 T^{12} + 316 p^{3} T^{13} + 16520 T^{14} + 33422 T^{15} + 66395 T^{16} + 99232 T^{17} + 64880 p T^{18} + 78548 p T^{19} + 540160 T^{20} + 78548 p^{2} T^{21} + 64880 p^{3} T^{22} + 99232 p^{3} T^{23} + 66395 p^{4} T^{24} + 33422 p^{5} T^{25} + 16520 p^{6} T^{26} + 316 p^{10} T^{27} + 4147 p^{8} T^{28} + 2660 p^{9} T^{29} + 1348 p^{10} T^{30} + 730 p^{11} T^{31} + 352 p^{12} T^{32} + 14 p^{15} T^{33} + 76 p^{14} T^{34} + 20 p^{15} T^{35} + 25 p^{16} T^{36} + 8 p^{17} T^{37} + 8 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40}$$
7 $$1 + 6 T + 62 T^{2} + 300 T^{3} + 1825 T^{4} + 7368 T^{5} + 33830 T^{6} + 117006 T^{7} + 444968 T^{8} + 1346382 T^{9} + 4485494 T^{10} + 12177600 T^{11} + 37478115 T^{12} + 13595244 p T^{13} + 286994378 T^{14} + 719113806 T^{15} + 2202916339 T^{16} + 5575023960 T^{17} + 17026952396 T^{18} + 42420168192 T^{19} + 124593483568 T^{20} + 42420168192 p T^{21} + 17026952396 p^{2} T^{22} + 5575023960 p^{3} T^{23} + 2202916339 p^{4} T^{24} + 719113806 p^{5} T^{25} + 286994378 p^{6} T^{26} + 13595244 p^{8} T^{27} + 37478115 p^{8} T^{28} + 12177600 p^{9} T^{29} + 4485494 p^{10} T^{30} + 1346382 p^{11} T^{31} + 444968 p^{12} T^{32} + 117006 p^{13} T^{33} + 33830 p^{14} T^{34} + 7368 p^{15} T^{35} + 1825 p^{16} T^{36} + 300 p^{17} T^{37} + 62 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40}$$
11 $$1 + 16 T + 140 T^{2} + 888 T^{3} + 4581 T^{4} + 20232 T^{5} + 78492 T^{6} + 266032 T^{7} + 70212 p T^{8} + 1836640 T^{9} + 3047260 T^{10} - 144872 T^{11} - 28959109 T^{12} - 162089992 T^{13} - 665171868 T^{14} - 2473230160 T^{15} - 9110751105 T^{16} - 33941309200 T^{17} - 124740924712 T^{18} - 443369710480 T^{19} - 1505615976312 T^{20} - 443369710480 p T^{21} - 124740924712 p^{2} T^{22} - 33941309200 p^{3} T^{23} - 9110751105 p^{4} T^{24} - 2473230160 p^{5} T^{25} - 665171868 p^{6} T^{26} - 162089992 p^{7} T^{27} - 28959109 p^{8} T^{28} - 144872 p^{9} T^{29} + 3047260 p^{10} T^{30} + 1836640 p^{11} T^{31} + 70212 p^{13} T^{32} + 266032 p^{13} T^{33} + 78492 p^{14} T^{34} + 20232 p^{15} T^{35} + 4581 p^{16} T^{36} + 888 p^{17} T^{37} + 140 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40}$$
17 $$1 + 10 T - 7 T^{2} - 470 T^{3} - 1882 T^{4} + 2042 T^{5} + 40575 T^{6} + 177466 T^{7} + 251190 T^{8} - 2495842 T^{9} - 12749509 T^{10} + 18741814 T^{11} + 217405948 T^{12} - 251724858 T^{13} - 4973586989 T^{14} - 12867053474 T^{15} + 23809984513 T^{16} + 417918910624 T^{17} + 1648824925642 T^{18} - 3203645045440 T^{19} - 42570382777428 T^{20} - 3203645045440 p T^{21} + 1648824925642 p^{2} T^{22} + 417918910624 p^{3} T^{23} + 23809984513 p^{4} T^{24} - 12867053474 p^{5} T^{25} - 4973586989 p^{6} T^{26} - 251724858 p^{7} T^{27} + 217405948 p^{8} T^{28} + 18741814 p^{9} T^{29} - 12749509 p^{10} T^{30} - 2495842 p^{11} T^{31} + 251190 p^{12} T^{32} + 177466 p^{13} T^{33} + 40575 p^{14} T^{34} + 2042 p^{15} T^{35} - 1882 p^{16} T^{36} - 470 p^{17} T^{37} - 7 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40}$$
19 $$1 - 20 T + 8 p T^{2} - 364 T^{3} - 1823 T^{4} + 13572 T^{5} - 15744 T^{6} - 58980 T^{7} - 559020 T^{8} + 6484036 T^{9} - 18760472 T^{10} - 16033220 T^{11} + 127241991 T^{12} + 326018372 T^{13} + 1166119272 T^{14} - 35902407492 T^{15} + 114702291975 T^{16} + 403614272520 T^{17} - 2514081037864 T^{18} - 7754137461800 T^{19} + 90038724597720 T^{20} - 7754137461800 p T^{21} - 2514081037864 p^{2} T^{22} + 403614272520 p^{3} T^{23} + 114702291975 p^{4} T^{24} - 35902407492 p^{5} T^{25} + 1166119272 p^{6} T^{26} + 326018372 p^{7} T^{27} + 127241991 p^{8} T^{28} - 16033220 p^{9} T^{29} - 18760472 p^{10} T^{30} + 6484036 p^{11} T^{31} - 559020 p^{12} T^{32} - 58980 p^{13} T^{33} - 15744 p^{14} T^{34} + 13572 p^{15} T^{35} - 1823 p^{16} T^{36} - 364 p^{17} T^{37} + 8 p^{19} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40}$$
23 $$1 + 2 T - 52 T^{2} - 160 T^{3} + 157 T^{4} + 6596 T^{5} + 46576 T^{6} - 124218 T^{7} - 1185616 T^{8} - 2300286 T^{9} + 6454608 T^{10} + 158042596 T^{11} + 339796655 T^{12} - 40412020 p T^{13} - 16319998268 T^{14} - 85172481666 T^{15} + 252762307659 T^{16} + 1967344793632 T^{17} + 8221712690560 T^{18} - 12847528296684 T^{19} - 399858827628576 T^{20} - 12847528296684 p T^{21} + 8221712690560 p^{2} T^{22} + 1967344793632 p^{3} T^{23} + 252762307659 p^{4} T^{24} - 85172481666 p^{5} T^{25} - 16319998268 p^{6} T^{26} - 40412020 p^{8} T^{27} + 339796655 p^{8} T^{28} + 158042596 p^{9} T^{29} + 6454608 p^{10} T^{30} - 2300286 p^{11} T^{31} - 1185616 p^{12} T^{32} - 124218 p^{13} T^{33} + 46576 p^{14} T^{34} + 6596 p^{15} T^{35} + 157 p^{16} T^{36} - 160 p^{17} T^{37} - 52 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40}$$
29 $$1 + 117 T^{2} + 6870 T^{4} + 8328 T^{5} + 9015 p T^{6} + 908016 T^{7} + 6368106 T^{8} + 49449240 T^{9} + 86554363 T^{10} + 1654451928 T^{11} - 517414740 T^{12} + 27891514632 T^{13} - 83953593117 T^{14} - 350011028904 T^{15} - 3978002845539 T^{16} - 1543536257664 p T^{17} - 154268740846974 T^{18} - 1909124569268712 T^{19} - 4952841504137748 T^{20} - 1909124569268712 p T^{21} - 154268740846974 p^{2} T^{22} - 1543536257664 p^{4} T^{23} - 3978002845539 p^{4} T^{24} - 350011028904 p^{5} T^{25} - 83953593117 p^{6} T^{26} + 27891514632 p^{7} T^{27} - 517414740 p^{8} T^{28} + 1654451928 p^{9} T^{29} + 86554363 p^{10} T^{30} + 49449240 p^{11} T^{31} + 6368106 p^{12} T^{32} + 908016 p^{13} T^{33} + 9015 p^{15} T^{34} + 8328 p^{15} T^{35} + 6870 p^{16} T^{36} + 117 p^{18} T^{38} + p^{20} T^{40}$$
31 $$1 - 104 T^{3} + 1794 T^{4} - 3320 T^{5} + 5408 T^{6} - 552640 T^{7} + 1315037 T^{8} + 1490240 T^{9} + 53283808 T^{10} - 779482432 T^{11} + 2015402168 T^{12} + 9179164672 T^{13} + 126121189280 T^{14} - 557942017984 T^{15} - 69210310750 T^{16} - 4465408437184 T^{17} + 158380681056 p^{2} T^{18} - 15988767100848 p T^{19} - 2929717194662260 T^{20} - 15988767100848 p^{2} T^{21} + 158380681056 p^{4} T^{22} - 4465408437184 p^{3} T^{23} - 69210310750 p^{4} T^{24} - 557942017984 p^{5} T^{25} + 126121189280 p^{6} T^{26} + 9179164672 p^{7} T^{27} + 2015402168 p^{8} T^{28} - 779482432 p^{9} T^{29} + 53283808 p^{10} T^{30} + 1490240 p^{11} T^{31} + 1315037 p^{12} T^{32} - 552640 p^{13} T^{33} + 5408 p^{14} T^{34} - 3320 p^{15} T^{35} + 1794 p^{16} T^{36} - 104 p^{17} T^{37} + p^{20} T^{40}$$
37 $$1 - 42 T + 1145 T^{2} - 23394 T^{3} + 397130 T^{4} - 5807850 T^{5} + 75530503 T^{6} - 887806170 T^{7} + 9573570974 T^{8} - 95583377406 T^{9} + 891367915243 T^{10} - 7811908786302 T^{11} + 64736532592752 T^{12} - 509675769422430 T^{13} + 3831380563778779 T^{14} - 27613939069692174 T^{15} + 191653487256049033 T^{16} - 34744221751356432 p T^{17} + 8363303716391015706 T^{18} - 1429745521339125912 p T^{19} +$$$$32\!\cdots\!64$$$$T^{20} - 1429745521339125912 p^{2} T^{21} + 8363303716391015706 p^{2} T^{22} - 34744221751356432 p^{4} T^{23} + 191653487256049033 p^{4} T^{24} - 27613939069692174 p^{5} T^{25} + 3831380563778779 p^{6} T^{26} - 509675769422430 p^{7} T^{27} + 64736532592752 p^{8} T^{28} - 7811908786302 p^{9} T^{29} + 891367915243 p^{10} T^{30} - 95583377406 p^{11} T^{31} + 9573570974 p^{12} T^{32} - 887806170 p^{13} T^{33} + 75530503 p^{14} T^{34} - 5807850 p^{15} T^{35} + 397130 p^{16} T^{36} - 23394 p^{17} T^{37} + 1145 p^{18} T^{38} - 42 p^{19} T^{39} + p^{20} T^{40}$$
41 $$1 - 10 T + 53 T^{2} + 370 T^{3} - 5474 T^{4} + 29582 T^{5} + 60499 T^{6} - 2473998 T^{7} + 17915882 T^{8} - 56078982 T^{9} - 425519157 T^{10} + 5305982318 T^{11} - 21403552108 T^{12} - 67497771986 T^{13} + 1513554719563 T^{14} - 7387733690454 T^{15} + 7177932842429 T^{16} + 156654968651636 T^{17} - 19723383509582 p T^{18} - 3491954639048348 T^{19} + 39411740829335628 T^{20} - 3491954639048348 p T^{21} - 19723383509582 p^{3} T^{22} + 156654968651636 p^{3} T^{23} + 7177932842429 p^{4} T^{24} - 7387733690454 p^{5} T^{25} + 1513554719563 p^{6} T^{26} - 67497771986 p^{7} T^{27} - 21403552108 p^{8} T^{28} + 5305982318 p^{9} T^{29} - 425519157 p^{10} T^{30} - 56078982 p^{11} T^{31} + 17915882 p^{12} T^{32} - 2473998 p^{13} T^{33} + 60499 p^{14} T^{34} + 29582 p^{15} T^{35} - 5474 p^{16} T^{36} + 370 p^{17} T^{37} + 53 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40}$$
43 $$1 + 22 T + 332 T^{2} + 3124 T^{3} + 24073 T^{4} + 137704 T^{5} + 827200 T^{6} + 115270 p T^{7} + 40616992 T^{8} + 296972046 T^{9} + 2336560152 T^{10} + 16772667664 T^{11} + 143141988371 T^{12} + 1164757348680 T^{13} + 9499206282004 T^{14} + 64461164121898 T^{15} + 415971606249867 T^{16} + 2327476919488472 T^{17} + 14227460893269592 T^{18} + 81794197398086924 T^{19} + 556966076767085568 T^{20} + 81794197398086924 p T^{21} + 14227460893269592 p^{2} T^{22} + 2327476919488472 p^{3} T^{23} + 415971606249867 p^{4} T^{24} + 64461164121898 p^{5} T^{25} + 9499206282004 p^{6} T^{26} + 1164757348680 p^{7} T^{27} + 143141988371 p^{8} T^{28} + 16772667664 p^{9} T^{29} + 2336560152 p^{10} T^{30} + 296972046 p^{11} T^{31} + 40616992 p^{12} T^{32} + 115270 p^{14} T^{33} + 827200 p^{14} T^{34} + 137704 p^{15} T^{35} + 24073 p^{16} T^{36} + 3124 p^{17} T^{37} + 332 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40}$$
47 $$1 - 572 T^{2} + 161342 T^{4} - 30069852 T^{6} + 4176817325 T^{8} - 461218805808 T^{10} + 895219768856 p T^{12} - 3248720168760816 T^{14} + 215624745949324178 T^{16} - 12422061007474931400 T^{18} +$$$$62\!\cdots\!44$$$$T^{20} - 12422061007474931400 p^{2} T^{22} + 215624745949324178 p^{4} T^{24} - 3248720168760816 p^{6} T^{26} + 895219768856 p^{9} T^{28} - 461218805808 p^{10} T^{30} + 4176817325 p^{12} T^{32} - 30069852 p^{14} T^{34} + 161342 p^{16} T^{36} - 572 p^{18} T^{38} + p^{20} T^{40}$$
53 $$1 + 10 T + 50 T^{2} + 608 T^{3} + 8911 T^{4} + 42256 T^{5} + 161842 T^{6} + 1379418 T^{7} + 11864269 T^{8} + 42078400 T^{9} + 116867400 T^{10} - 342809968 T^{11} + 2993373476 T^{12} + 82242842224 T^{13} + 232250742760 T^{14} - 4408284842096 T^{15} - 104445179440542 T^{16} - 7400825808492 p T^{17} - 1433330906377060 T^{18} - 34029241052501104 T^{19} - 545288959843663782 T^{20} - 34029241052501104 p T^{21} - 1433330906377060 p^{2} T^{22} - 7400825808492 p^{4} T^{23} - 104445179440542 p^{4} T^{24} - 4408284842096 p^{5} T^{25} + 232250742760 p^{6} T^{26} + 82242842224 p^{7} T^{27} + 2993373476 p^{8} T^{28} - 342809968 p^{9} T^{29} + 116867400 p^{10} T^{30} + 42078400 p^{11} T^{31} + 11864269 p^{12} T^{32} + 1379418 p^{13} T^{33} + 161842 p^{14} T^{34} + 42256 p^{15} T^{35} + 8911 p^{16} T^{36} + 608 p^{17} T^{37} + 50 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40}$$
59 $$1 - 16 T + 320 T^{2} - 3664 T^{3} + 37117 T^{4} - 331488 T^{5} + 2103744 T^{6} - 16380920 T^{7} + 100348484 T^{8} - 883337776 T^{9} + 9675056288 T^{10} - 81687354080 T^{11} + 809702972683 T^{12} - 5715721431720 T^{13} + 35547465831840 T^{14} - 241780218876136 T^{15} + 1057616949186407 T^{16} - 10092819361240512 T^{17} + 89215361760986816 T^{18} - 721667655155172056 T^{19} + 7602396017918841272 T^{20} - 721667655155172056 p T^{21} + 89215361760986816 p^{2} T^{22} - 10092819361240512 p^{3} T^{23} + 1057616949186407 p^{4} T^{24} - 241780218876136 p^{5} T^{25} + 35547465831840 p^{6} T^{26} - 5715721431720 p^{7} T^{27} + 809702972683 p^{8} T^{28} - 81687354080 p^{9} T^{29} + 9675056288 p^{10} T^{30} - 883337776 p^{11} T^{31} + 100348484 p^{12} T^{32} - 16380920 p^{13} T^{33} + 2103744 p^{14} T^{34} - 331488 p^{15} T^{35} + 37117 p^{16} T^{36} - 3664 p^{17} T^{37} + 320 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40}$$
61 $$1 + 16 T - 291 T^{2} - 4624 T^{3} + 62614 T^{4} + 791392 T^{5} - 10200029 T^{6} - 91274480 T^{7} + 1347937546 T^{8} + 7746824480 T^{9} - 143223609453 T^{10} - 467039401632 T^{11} + 12554800039532 T^{12} + 18247567668448 T^{13} - 918360371027813 T^{14} - 202692877277696 T^{15} + 58440043341660413 T^{16} - 20244855642251520 T^{17} - 3426930072683677342 T^{18} + 853407459595356016 T^{19} +$$$$20\!\cdots\!68$$$$T^{20} + 853407459595356016 p T^{21} - 3426930072683677342 p^{2} T^{22} - 20244855642251520 p^{3} T^{23} + 58440043341660413 p^{4} T^{24} - 202692877277696 p^{5} T^{25} - 918360371027813 p^{6} T^{26} + 18247567668448 p^{7} T^{27} + 12554800039532 p^{8} T^{28} - 467039401632 p^{9} T^{29} - 143223609453 p^{10} T^{30} + 7746824480 p^{11} T^{31} + 1347937546 p^{12} T^{32} - 91274480 p^{13} T^{33} - 10200029 p^{14} T^{34} + 791392 p^{15} T^{35} + 62614 p^{16} T^{36} - 4624 p^{17} T^{37} - 291 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40}$$
67 $$1 + 58 T + 1374 T^{2} + 16980 T^{3} + 138265 T^{4} + 1570168 T^{5} + 24256358 T^{6} + 249069002 T^{7} + 1682109232 T^{8} + 14296545234 T^{9} + 163570728638 T^{10} + 1243351040112 T^{11} + 6261994153619 T^{12} + 48928472730988 T^{13} + 414692298036122 T^{14} + 727003409615386 T^{15} - 13677815969660389 T^{16} - 77464561742949576 T^{17} - 533738706610505228 T^{18} - 14605719503956384576 T^{19} -$$$$17\!\cdots\!40$$$$T^{20} - 14605719503956384576 p T^{21} - 533738706610505228 p^{2} T^{22} - 77464561742949576 p^{3} T^{23} - 13677815969660389 p^{4} T^{24} + 727003409615386 p^{5} T^{25} + 414692298036122 p^{6} T^{26} + 48928472730988 p^{7} T^{27} + 6261994153619 p^{8} T^{28} + 1243351040112 p^{9} T^{29} + 163570728638 p^{10} T^{30} + 14296545234 p^{11} T^{31} + 1682109232 p^{12} T^{32} + 249069002 p^{13} T^{33} + 24256358 p^{14} T^{34} + 1570168 p^{15} T^{35} + 138265 p^{16} T^{36} + 16980 p^{17} T^{37} + 1374 p^{18} T^{38} + 58 p^{19} T^{39} + p^{20} T^{40}$$
71 $$1 + 16 T - 256 T^{2} - 3084 T^{3} + 66425 T^{4} + 512092 T^{5} - 8625144 T^{6} - 36645360 T^{7} + 960694980 T^{8} + 3380563112 T^{9} - 58860816752 T^{10} - 149366684012 T^{11} + 3433021756335 T^{12} + 19056716386500 T^{13} + 18952756972224 T^{14} - 524932892465792 T^{15} - 10473823723399401 T^{16} + 71341917858912984 T^{17} + 2184336262642299176 T^{18} + 1327381202060222744 T^{19} -$$$$14\!\cdots\!00$$$$T^{20} + 1327381202060222744 p T^{21} + 2184336262642299176 p^{2} T^{22} + 71341917858912984 p^{3} T^{23} - 10473823723399401 p^{4} T^{24} - 524932892465792 p^{5} T^{25} + 18952756972224 p^{6} T^{26} + 19056716386500 p^{7} T^{27} + 3433021756335 p^{8} T^{28} - 149366684012 p^{9} T^{29} - 58860816752 p^{10} T^{30} + 3380563112 p^{11} T^{31} + 960694980 p^{12} T^{32} - 36645360 p^{13} T^{33} - 8625144 p^{14} T^{34} + 512092 p^{15} T^{35} + 66425 p^{16} T^{36} - 3084 p^{17} T^{37} - 256 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40}$$
73 $$( 1 - 36 T + 979 T^{2} - 17720 T^{3} + 269289 T^{4} - 3223420 T^{5} + 34128052 T^{6} - 304511620 T^{7} + 2578848294 T^{8} - 20162242116 T^{9} + 171086571074 T^{10} - 20162242116 p T^{11} + 2578848294 p^{2} T^{12} - 304511620 p^{3} T^{13} + 34128052 p^{4} T^{14} - 3223420 p^{5} T^{15} + 269289 p^{6} T^{16} - 17720 p^{7} T^{17} + 979 p^{8} T^{18} - 36 p^{9} T^{19} + p^{10} T^{20} )^{2}$$
79 $$1 - 772 T^{2} + 300974 T^{4} - 79250180 T^{6} + 15881528397 T^{8} - 2583290881840 T^{10} + 354721609143528 T^{12} - 42178231844928304 T^{14} + 4416281458244224626 T^{16} -$$$$41\!\cdots\!84$$$$T^{18} +$$$$34\!\cdots\!44$$$$T^{20} -$$$$41\!\cdots\!84$$$$p^{2} T^{22} + 4416281458244224626 p^{4} T^{24} - 42178231844928304 p^{6} T^{26} + 354721609143528 p^{8} T^{28} - 2583290881840 p^{10} T^{30} + 15881528397 p^{12} T^{32} - 79250180 p^{14} T^{34} + 300974 p^{16} T^{36} - 772 p^{18} T^{38} + p^{20} T^{40}$$
83 $$1 - 1212 T^{2} + 716166 T^{4} - 274705788 T^{6} + 76836687021 T^{8} - 16688569970480 T^{10} + 2926110414583464 T^{12} - 424954841088102384 T^{14} + 52013437736047347810 T^{16} -$$$$54\!\cdots\!00$$$$T^{18} +$$$$48\!\cdots\!20$$$$T^{20} -$$$$54\!\cdots\!00$$$$p^{2} T^{22} + 52013437736047347810 p^{4} T^{24} - 424954841088102384 p^{6} T^{26} + 2926110414583464 p^{8} T^{28} - 16688569970480 p^{10} T^{30} + 76836687021 p^{12} T^{32} - 274705788 p^{14} T^{34} + 716166 p^{16} T^{36} - 1212 p^{18} T^{38} + p^{20} T^{40}$$
89 $$1 - 6 T - 150 T^{2} - 808 T^{3} + 3685 T^{4} + 327028 T^{5} + 1849958 T^{6} - 16865574 T^{7} - 355208492 T^{8} - 3406428486 T^{9} + 23770368054 T^{10} + 554018688884 T^{11} + 2507154486755 T^{12} - 21679381682636 T^{13} - 586967110638826 T^{14} - 35926910654726 p T^{15} + 29570324373338103 T^{16} + 519121278252789240 T^{17} + 2865486495313106852 T^{18} - 22835320674508257412 T^{19} -$$$$51\!\cdots\!48$$$$T^{20} - 22835320674508257412 p T^{21} + 2865486495313106852 p^{2} T^{22} + 519121278252789240 p^{3} T^{23} + 29570324373338103 p^{4} T^{24} - 35926910654726 p^{6} T^{25} - 586967110638826 p^{6} T^{26} - 21679381682636 p^{7} T^{27} + 2507154486755 p^{8} T^{28} + 554018688884 p^{9} T^{29} + 23770368054 p^{10} T^{30} - 3406428486 p^{11} T^{31} - 355208492 p^{12} T^{32} - 16865574 p^{13} T^{33} + 1849958 p^{14} T^{34} + 327028 p^{15} T^{35} + 3685 p^{16} T^{36} - 808 p^{17} T^{37} - 150 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40}$$
97 $$1 + 22 T - 174 T^{2} - 6084 T^{3} + 20569 T^{4} + 989140 T^{5} - 1173910 T^{6} - 117668482 T^{7} - 323459672 T^{8} + 9141570042 T^{9} + 92511246554 T^{10} - 309361777500 T^{11} - 12599002237573 T^{12} - 32546855510684 T^{13} + 1080345492224582 T^{14} + 7575934089129994 T^{15} - 46719362005426429 T^{16} - 767937641520707160 T^{17} - 1640677573678341644 T^{18} + 32908917429876961448 T^{19} +$$$$43\!\cdots\!88$$$$T^{20} + 32908917429876961448 p T^{21} - 1640677573678341644 p^{2} T^{22} - 767937641520707160 p^{3} T^{23} - 46719362005426429 p^{4} T^{24} + 7575934089129994 p^{5} T^{25} + 1080345492224582 p^{6} T^{26} - 32546855510684 p^{7} T^{27} - 12599002237573 p^{8} T^{28} - 309361777500 p^{9} T^{29} + 92511246554 p^{10} T^{30} + 9141570042 p^{11} T^{31} - 323459672 p^{12} T^{32} - 117668482 p^{13} T^{33} - 1173910 p^{14} T^{34} + 989140 p^{15} T^{35} + 20569 p^{16} T^{36} - 6084 p^{17} T^{37} - 174 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$