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 + 8·9-s + 24·10-s − 8·11-s − 30·12-s − 24·14-s − 12·15-s + 99·16-s − 4·17-s + 32·18-s + 16·19-s + 90·20-s + 12·21-s − 32·22-s + 10·23-s − 80·24-s + 9·25-s − 8·27-s − 90·28-s − 48·30-s + 210·32-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 + 8/3·9-s + 7.58·10-s − 2.41·11-s − 8.66·12-s − 6.41·14-s − 3.09·15-s + 99/4·16-s − 0.970·17-s + 7.54·18-s + 3.67·19-s + 20.1·20-s + 2.61·21-s − 6.82·22-s + 2.08·23-s − 16.3·24-s + 9/5·25-s − 1.53·27-s − 17.0·28-s − 8.76·30-s + 37.1·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 13^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.82539\times 10^{16}\) |
| Root analytic conductor: | \(2.59756\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{20} \cdot 13^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(142.6692763\) |
| \(L(\frac12)\) | \(\approx\) | \(142.6692763\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 6 T + 27 T^{2} - 114 T^{3} + 393 T^{4} - 1244 T^{5} + 3672 T^{6} - 10116 T^{7} + 25998 T^{8} - 63264 T^{9} + 146954 T^{10} - 63264 p T^{11} + 25998 p^{2} T^{12} - 10116 p^{3} T^{13} + 3672 p^{4} T^{14} - 1244 p^{5} T^{15} + 393 p^{6} T^{16} - 114 p^{7} T^{17} + 27 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T + T^{2} + p^{4} T^{3} - 9 p T^{4} - 11 p T^{5} + 45 T^{6} - 5 p T^{7} - 15 T^{8} + 11 p^{2} T^{9} - 9 p^{4} T^{10} + 45 p^{2} T^{11} + 41 T^{12} - 151 p^{2} T^{13} + 1081 T^{14} + p^{5} T^{15} - 1375 p T^{16} + 1397 p T^{17} + 1609 T^{18} - 1915 p T^{19} + 2909 T^{20} - 1915 p^{2} T^{21} + 1609 p^{2} T^{22} + 1397 p^{4} T^{23} - 1375 p^{5} T^{24} + p^{10} T^{25} + 1081 p^{6} T^{26} - 151 p^{9} T^{27} + 41 p^{8} T^{28} + 45 p^{11} T^{29} - 9 p^{14} T^{30} + 11 p^{13} T^{31} - 15 p^{12} T^{32} - 5 p^{14} T^{33} + 45 p^{14} T^{34} - 11 p^{16} T^{35} - 9 p^{17} T^{36} + p^{21} T^{37} + p^{18} T^{38} - p^{21} T^{39} + p^{20} T^{40} \) |
| 3 | \( 1 + 2 T - 4 T^{2} - 16 T^{3} + T^{4} + 56 T^{5} + 40 T^{6} - 50 p T^{7} - 176 T^{8} + 406 T^{9} + 640 T^{10} - 1360 T^{11} - 2741 T^{12} + 1196 p T^{13} + 9092 T^{14} - 11098 T^{15} - 32293 T^{16} + 42280 T^{17} + 47704 p T^{18} - 19340 p T^{19} - 519968 T^{20} - 19340 p^{2} T^{21} + 47704 p^{3} T^{22} + 42280 p^{3} T^{23} - 32293 p^{4} T^{24} - 11098 p^{5} T^{25} + 9092 p^{6} T^{26} + 1196 p^{8} T^{27} - 2741 p^{8} T^{28} - 1360 p^{9} T^{29} + 640 p^{10} T^{30} + 406 p^{11} T^{31} - 176 p^{12} T^{32} - 50 p^{14} T^{33} + 40 p^{14} T^{34} + 56 p^{15} T^{35} + p^{16} T^{36} - 16 p^{17} T^{37} - 4 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 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 | \( 1 + 8 T + 20 T^{2} - 24 T^{3} - 219 T^{4} - 888 T^{5} - 4284 T^{6} - 15880 T^{7} - 12564 T^{8} + 158360 T^{9} + 899332 T^{10} + 2756696 T^{11} + 7669307 T^{12} + 25336 p^{2} T^{13} - 105036036 T^{14} - 513543080 T^{15} - 1219479681 T^{16} - 2337775184 T^{17} + 1553351912 T^{18} + 44546706496 T^{19} + 220101622920 T^{20} + 44546706496 p T^{21} + 1553351912 p^{2} T^{22} - 2337775184 p^{3} T^{23} - 1219479681 p^{4} T^{24} - 513543080 p^{5} T^{25} - 105036036 p^{6} T^{26} + 25336 p^{9} T^{27} + 7669307 p^{8} T^{28} + 2756696 p^{9} T^{29} + 899332 p^{10} T^{30} + 158360 p^{11} T^{31} - 12564 p^{12} T^{32} - 15880 p^{13} T^{33} - 4284 p^{14} T^{34} - 888 p^{15} T^{35} - 219 p^{16} T^{36} - 24 p^{17} T^{37} + 20 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 4 T + 65 T^{2} + 172 T^{3} + 1418 T^{4} + 2228 T^{5} + 8655 T^{6} - 1808 T^{7} - 268746 T^{8} - 1272808 T^{9} - 13137085 T^{10} - 51346880 T^{11} - 296572088 T^{12} - 838147728 T^{13} - 3153368693 T^{14} - 3609124160 T^{15} + 11120075953 T^{16} + 145288002868 T^{17} + 1472996590450 T^{18} + 5409253581872 T^{19} + 36582230959020 T^{20} + 5409253581872 p T^{21} + 1472996590450 p^{2} T^{22} + 145288002868 p^{3} T^{23} + 11120075953 p^{4} T^{24} - 3609124160 p^{5} T^{25} - 3153368693 p^{6} T^{26} - 838147728 p^{7} T^{27} - 296572088 p^{8} T^{28} - 51346880 p^{9} T^{29} - 13137085 p^{10} T^{30} - 1272808 p^{11} T^{31} - 268746 p^{12} T^{32} - 1808 p^{13} T^{33} + 8655 p^{14} T^{34} + 2228 p^{15} T^{35} + 1418 p^{16} T^{36} + 172 p^{17} T^{37} + 65 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 16 T + 176 T^{2} - 1460 T^{3} + 10321 T^{4} - 63732 T^{5} + 356664 T^{6} - 1829664 T^{7} + 8759124 T^{8} - 39787552 T^{9} + 174977008 T^{10} - 761420092 T^{11} + 3335898519 T^{12} - 14694366620 T^{13} + 64786527696 T^{14} - 282753607560 T^{15} + 1213406803719 T^{16} - 5111934152664 T^{17} + 21227495686040 T^{18} - 88894314395488 T^{19} + 379894945670616 T^{20} - 88894314395488 p T^{21} + 21227495686040 p^{2} T^{22} - 5111934152664 p^{3} T^{23} + 1213406803719 p^{4} T^{24} - 282753607560 p^{5} T^{25} + 64786527696 p^{6} T^{26} - 14694366620 p^{7} T^{27} + 3335898519 p^{8} T^{28} - 761420092 p^{9} T^{29} + 174977008 p^{10} T^{30} - 39787552 p^{11} T^{31} + 8759124 p^{12} T^{32} - 1829664 p^{13} T^{33} + 356664 p^{14} T^{34} - 63732 p^{15} T^{35} + 10321 p^{16} T^{36} - 1460 p^{17} T^{37} + 176 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 10 T + 104 T^{2} - 760 T^{3} + 4021 T^{4} - 25240 T^{5} + 110596 T^{6} - 629514 T^{7} + 3599888 T^{8} - 16216374 T^{9} + 101637156 T^{10} - 372526544 T^{11} + 1815379751 T^{12} - 8866821764 T^{13} + 37211800024 T^{14} - 311193036366 T^{15} + 1423456049115 T^{16} - 8723716465064 T^{17} + 42970194226792 T^{18} - 170399078825124 T^{19} + 1023524571922080 T^{20} - 170399078825124 p T^{21} + 42970194226792 p^{2} T^{22} - 8723716465064 p^{3} T^{23} + 1423456049115 p^{4} T^{24} - 311193036366 p^{5} T^{25} + 37211800024 p^{6} T^{26} - 8866821764 p^{7} T^{27} + 1815379751 p^{8} T^{28} - 372526544 p^{9} T^{29} + 101637156 p^{10} T^{30} - 16216374 p^{11} T^{31} + 3599888 p^{12} T^{32} - 629514 p^{13} T^{33} + 110596 p^{14} T^{34} - 25240 p^{15} T^{35} + 4021 p^{16} T^{36} - 760 p^{17} T^{37} + 104 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 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 | \( 1 + 28 T + 389 T^{2} + 3008 T^{3} + 8194 T^{4} - 92960 T^{5} - 1264061 T^{6} - 6519924 T^{7} - 3613294 T^{8} + 159299460 T^{9} + 822628347 T^{10} - 787403984 T^{11} - 19305288280 T^{12} + 60662487584 T^{13} + 2225226104059 T^{14} + 20587364409156 T^{15} + 114095998027205 T^{16} + 292309852251568 T^{17} - 1972676048100766 T^{18} - 34827701209580176 T^{19} - 281355382830851844 T^{20} - 34827701209580176 p T^{21} - 1972676048100766 p^{2} T^{22} + 292309852251568 p^{3} T^{23} + 114095998027205 p^{4} T^{24} + 20587364409156 p^{5} T^{25} + 2225226104059 p^{6} T^{26} + 60662487584 p^{7} T^{27} - 19305288280 p^{8} T^{28} - 787403984 p^{9} T^{29} + 822628347 p^{10} T^{30} + 159299460 p^{11} T^{31} - 3613294 p^{12} T^{32} - 6519924 p^{13} T^{33} - 1264061 p^{14} T^{34} - 92960 p^{15} T^{35} + 8194 p^{16} T^{36} + 3008 p^{17} T^{37} + 389 p^{18} T^{38} + 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 10 T - 40 T^{2} - 1364 T^{3} - 3983 T^{4} + 77956 T^{5} + 640228 T^{6} - 1596326 T^{7} - 38848928 T^{8} - 76430850 T^{9} + 1212163500 T^{10} + 6607056052 T^{11} - 135285871 p T^{12} - 76098476592 T^{13} - 657626537288 T^{14} - 14318581328546 T^{15} - 47723084208453 T^{16} + 1005173616928256 T^{17} + 8200326933037552 T^{18} - 21221387803082188 T^{19} - 490364809303665792 T^{20} - 21221387803082188 p T^{21} + 8200326933037552 p^{2} T^{22} + 1005173616928256 p^{3} T^{23} - 47723084208453 p^{4} T^{24} - 14318581328546 p^{5} T^{25} - 657626537288 p^{6} T^{26} - 76098476592 p^{7} T^{27} - 135285871 p^{9} T^{28} + 6607056052 p^{9} T^{29} + 1212163500 p^{10} T^{30} - 76430850 p^{11} T^{31} - 38848928 p^{12} T^{32} - 1596326 p^{13} T^{33} + 640228 p^{14} T^{34} + 77956 p^{15} T^{35} - 3983 p^{16} T^{36} - 1364 p^{17} T^{37} - 40 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 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 | \( 1 - 8 T - 160 T^{2} + 1360 T^{3} + 6013 T^{4} - 102072 T^{5} + 738528 T^{6} + 2533208 T^{7} - 73281532 T^{8} + 209404528 T^{9} + 37464416 T^{10} - 26337573400 T^{11} + 280444882123 T^{12} + 1108551577128 T^{13} - 11218760651904 T^{14} + 24021446097136 T^{15} - 143672083299481 T^{16} - 5273614558649664 T^{17} + 5977156174031072 T^{18} + 175413463692198584 T^{19} + 707142863345228216 T^{20} + 175413463692198584 p T^{21} + 5977156174031072 p^{2} T^{22} - 5273614558649664 p^{3} T^{23} - 143672083299481 p^{4} T^{24} + 24021446097136 p^{5} T^{25} - 11218760651904 p^{6} T^{26} + 1108551577128 p^{7} T^{27} + 280444882123 p^{8} T^{28} - 26337573400 p^{9} T^{29} + 37464416 p^{10} T^{30} + 209404528 p^{11} T^{31} - 73281532 p^{12} T^{32} + 2533208 p^{13} T^{33} + 738528 p^{14} T^{34} - 102072 p^{15} T^{35} + 6013 p^{16} T^{36} + 1360 p^{17} T^{37} - 160 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 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 | \( 1 + 56 T + 1952 T^{2} + 48396 T^{3} + 953561 T^{4} + 15523292 T^{5} + 217618632 T^{6} + 2711712552 T^{7} + 31230772164 T^{8} + 345744549544 T^{9} + 3784871417104 T^{10} + 41211074532692 T^{11} + 438822748561071 T^{12} + 4488068016117828 T^{13} + 43707379067808384 T^{14} + 408784301786439440 T^{15} + 3737075509158186519 T^{16} + 33940694818139883288 T^{17} + \)\(30\!\cdots\!88\)\( T^{18} + \)\(27\!\cdots\!52\)\( T^{19} + \)\(23\!\cdots\!44\)\( T^{20} + \)\(27\!\cdots\!52\)\( p T^{21} + \)\(30\!\cdots\!88\)\( p^{2} T^{22} + 33940694818139883288 p^{3} T^{23} + 3737075509158186519 p^{4} T^{24} + 408784301786439440 p^{5} T^{25} + 43707379067808384 p^{6} T^{26} + 4488068016117828 p^{7} T^{27} + 438822748561071 p^{8} T^{28} + 41211074532692 p^{9} T^{29} + 3784871417104 p^{10} T^{30} + 345744549544 p^{11} T^{31} + 31230772164 p^{12} T^{32} + 2711712552 p^{13} T^{33} + 217618632 p^{14} T^{34} + 15523292 p^{15} T^{35} + 953561 p^{16} T^{36} + 48396 p^{17} T^{37} + 1952 p^{18} T^{38} + 56 p^{19} T^{39} + p^{20} T^{40} \) | |
| 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 | \( 1 + 6 T + 186 T^{2} + 2824 T^{3} + 9733 T^{4} + 476660 T^{5} + 2104166 T^{6} + 32440014 T^{7} + 639120004 T^{8} + 1892373150 T^{9} + 80881921830 T^{10} + 470701967188 T^{11} + 5583842052083 T^{12} + 81722613898076 T^{13} + 320383685920070 T^{14} + 8669810929759270 T^{15} + 38773926433759431 T^{16} + 628346584296533016 T^{17} + 6870273527349716276 T^{18} + 34025577862123733188 T^{19} + \)\(80\!\cdots\!12\)\( T^{20} + 34025577862123733188 p T^{21} + 6870273527349716276 p^{2} T^{22} + 628346584296533016 p^{3} T^{23} + 38773926433759431 p^{4} T^{24} + 8669810929759270 p^{5} T^{25} + 320383685920070 p^{6} T^{26} + 81722613898076 p^{7} T^{27} + 5583842052083 p^{8} T^{28} + 470701967188 p^{9} T^{29} + 80881921830 p^{10} T^{30} + 1892373150 p^{11} T^{31} + 639120004 p^{12} T^{32} + 32440014 p^{13} T^{33} + 2104166 p^{14} T^{34} + 476660 p^{15} T^{35} + 9733 p^{16} T^{36} + 2824 p^{17} T^{37} + 186 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 22 T - 174 T^{2} + 6084 T^{3} + 20569 T^{4} - 989140 T^{5} - 1173910 T^{6} + 117668482 T^{7} - 323459672 T^{8} - 9141570042 T^{9} + 92511246554 T^{10} + 309361777500 T^{11} - 12599002237573 T^{12} + 32546855510684 T^{13} + 1080345492224582 T^{14} - 7575934089129994 T^{15} - 46719362005426429 T^{16} + 767937641520707160 T^{17} - 1640677573678341644 T^{18} - 32908917429876961448 T^{19} + \)\(43\!\cdots\!88\)\( T^{20} - 32908917429876961448 p T^{21} - 1640677573678341644 p^{2} T^{22} + 767937641520707160 p^{3} T^{23} - 46719362005426429 p^{4} T^{24} - 7575934089129994 p^{5} T^{25} + 1080345492224582 p^{6} T^{26} + 32546855510684 p^{7} T^{27} - 12599002237573 p^{8} T^{28} + 309361777500 p^{9} T^{29} + 92511246554 p^{10} T^{30} - 9141570042 p^{11} T^{31} - 323459672 p^{12} T^{32} + 117668482 p^{13} T^{33} - 1173910 p^{14} T^{34} - 989140 p^{15} T^{35} + 20569 p^{16} T^{36} + 6084 p^{17} T^{37} - 174 p^{18} T^{38} - 22 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.25890504968450601898180702488, −2.23547791299334040321388757579, −2.11708988784914787267184858395, −2.00457588243729505543616795913, −1.92967669044591834369199187161, −1.86473631977276811459285589520, −1.80280447777616222552214551631, −1.78213643688052248627814746204, −1.76252001817164562379434348016, −1.72583650053081802447716346691, −1.72457389597469940919036119248, −1.67620575315463795903694818170, −1.58588763845296692621616874732, −1.51228913714555973652314100732, −1.45846801223690050075524390472, −1.29153401544000360916509654711, −1.05839052707244215608694513541, −0.934797894276624070646443667056, −0.895754752050333829064064590425, −0.827955740471077355126066712508, −0.74365105386565660617043900097, −0.70840668876880470192964928534, −0.60124345112753304849274224708, −0.24591915397920695675431008948, −0.12990026807385655486915820812, 0.12990026807385655486915820812, 0.24591915397920695675431008948, 0.60124345112753304849274224708, 0.70840668876880470192964928534, 0.74365105386565660617043900097, 0.827955740471077355126066712508, 0.895754752050333829064064590425, 0.934797894276624070646443667056, 1.05839052707244215608694513541, 1.29153401544000360916509654711, 1.45846801223690050075524390472, 1.51228913714555973652314100732, 1.58588763845296692621616874732, 1.67620575315463795903694818170, 1.72457389597469940919036119248, 1.72583650053081802447716346691, 1.76252001817164562379434348016, 1.78213643688052248627814746204, 1.80280447777616222552214551631, 1.86473631977276811459285589520, 1.92967669044591834369199187161, 2.00457588243729505543616795913, 2.11708988784914787267184858395, 2.23547791299334040321388757579, 2.25890504968450601898180702488
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.