Dirichlet series
| L(s) = 1 | + 3-s + 5-s + 3·7-s + 14·9-s − 11-s − 5·13-s + 15-s − 3·17-s − 7·19-s + 3·21-s − 30·23-s + 7·25-s + 22·27-s − 14·29-s + 11·31-s − 33-s + 3·35-s + 45·37-s − 5·39-s + 9·41-s − 8·43-s + 14·45-s + 39·47-s + 37·49-s − 3·51-s + 36·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 14/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s − 0.727·17-s − 1.60·19-s + 0.654·21-s − 6.25·23-s + 7/5·25-s + 4.23·27-s − 2.59·29-s + 1.97·31-s − 0.174·33-s + 0.507·35-s + 7.39·37-s − 0.800·39-s + 1.40·41-s − 1.21·43-s + 2.08·45-s + 5.68·47-s + 37/7·49-s − 0.420·51-s + 4.94·53-s − 0.134·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 11^{20} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56121\times 10^{13}\) |
| Root analytic conductor: | \(2.13715\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 11^{20} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(94.56460779\) |
| \(L(\frac12)\) | \(\approx\) | \(94.56460779\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T + 28 T^{2} - 12 T^{3} + 376 T^{4} - 700 T^{5} + 274 p T^{6} - 15448 T^{7} + 10157 T^{8} - 202266 T^{9} + 38073 T^{10} - 202266 p T^{11} + 10157 p^{2} T^{12} - 15448 p^{3} T^{13} + 274 p^{5} T^{14} - 700 p^{5} T^{15} + 376 p^{6} T^{16} - 12 p^{7} T^{17} + 28 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) | |
| good | 3 | \( 1 - T - 13 T^{2} + 5 T^{3} + 79 T^{4} + 2 p^{3} T^{5} - 286 T^{6} - 223 p T^{7} + 380 T^{8} + 3178 T^{9} + 100 p^{3} T^{10} - 2258 p T^{11} - 20296 T^{12} - 3107 p T^{13} + 62165 T^{14} + 124814 T^{15} - 1159 p^{3} T^{16} - 441791 T^{17} - 573202 T^{18} + 195511 p T^{19} + 2675671 T^{20} + 195511 p^{2} T^{21} - 573202 p^{2} T^{22} - 441791 p^{3} T^{23} - 1159 p^{7} T^{24} + 124814 p^{5} T^{25} + 62165 p^{6} T^{26} - 3107 p^{8} T^{27} - 20296 p^{8} T^{28} - 2258 p^{10} T^{29} + 100 p^{13} T^{30} + 3178 p^{11} T^{31} + 380 p^{12} T^{32} - 223 p^{14} T^{33} - 286 p^{14} T^{34} + 2 p^{18} T^{35} + 79 p^{16} T^{36} + 5 p^{17} T^{37} - 13 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 - T - 6 T^{2} + 3 p T^{4} - 166 T^{5} + 306 T^{6} + 1056 T^{7} - 108 p T^{8} - 853 p T^{9} + 12389 T^{10} - 28534 T^{11} - 71039 T^{12} + 18934 p T^{13} + 81352 p T^{14} - 6562 p^{3} T^{15} + 57338 p^{2} T^{16} + 24246 p^{3} T^{17} - 52546 p^{3} T^{18} - 30186 p^{4} T^{19} + 3377 p^{6} T^{20} - 30186 p^{5} T^{21} - 52546 p^{5} T^{22} + 24246 p^{6} T^{23} + 57338 p^{6} T^{24} - 6562 p^{8} T^{25} + 81352 p^{7} T^{26} + 18934 p^{8} T^{27} - 71039 p^{8} T^{28} - 28534 p^{9} T^{29} + 12389 p^{10} T^{30} - 853 p^{12} T^{31} - 108 p^{13} T^{32} + 1056 p^{13} T^{33} + 306 p^{14} T^{34} - 166 p^{15} T^{35} + 3 p^{17} T^{36} - 6 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 3 T - 4 p T^{2} + 66 T^{3} + 388 T^{4} - 123 p T^{5} - 2252 T^{6} + 9088 T^{7} - 3809 T^{8} - 89183 T^{9} + 154069 T^{10} + 532561 T^{11} - 1261160 T^{12} - 197198 T^{13} + 8962717 T^{14} - 2576963 p T^{15} - 48416254 T^{16} + 88683698 T^{17} - 73463546 T^{18} - 92044627 T^{19} + 2453233351 T^{20} - 92044627 p T^{21} - 73463546 p^{2} T^{22} + 88683698 p^{3} T^{23} - 48416254 p^{4} T^{24} - 2576963 p^{6} T^{25} + 8962717 p^{6} T^{26} - 197198 p^{7} T^{27} - 1261160 p^{8} T^{28} + 532561 p^{9} T^{29} + 154069 p^{10} T^{30} - 89183 p^{11} T^{31} - 3809 p^{12} T^{32} + 9088 p^{13} T^{33} - 2252 p^{14} T^{34} - 123 p^{16} T^{35} + 388 p^{16} T^{36} + 66 p^{17} T^{37} - 4 p^{19} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 3 T - 79 T^{2} - 312 T^{3} + 2830 T^{4} + 16345 T^{5} - 53286 T^{6} - 555142 T^{7} + 121306 T^{8} + 12512854 T^{9} + 24761983 T^{10} - 160558830 T^{11} - 793089927 T^{12} - 118069684 T^{13} + 11270133834 T^{14} + 54380532735 T^{15} + 26614217831 T^{16} - 1178894039769 T^{17} - 5143253807945 T^{18} + 9063030233332 T^{19} + 124012151808064 T^{20} + 9063030233332 p T^{21} - 5143253807945 p^{2} T^{22} - 1178894039769 p^{3} T^{23} + 26614217831 p^{4} T^{24} + 54380532735 p^{5} T^{25} + 11270133834 p^{6} T^{26} - 118069684 p^{7} T^{27} - 793089927 p^{8} T^{28} - 160558830 p^{9} T^{29} + 24761983 p^{10} T^{30} + 12512854 p^{11} T^{31} + 121306 p^{12} T^{32} - 555142 p^{13} T^{33} - 53286 p^{14} T^{34} + 16345 p^{15} T^{35} + 2830 p^{16} T^{36} - 312 p^{17} T^{37} - 79 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 7 T - 36 T^{2} - 222 T^{3} + 90 p T^{4} + 7647 T^{5} - 48737 T^{6} - 192056 T^{7} + 1355255 T^{8} + 4946643 T^{9} - 27101700 T^{10} - 78317664 T^{11} + 558528114 T^{12} + 1551192102 T^{13} - 7858344544 T^{14} - 20888923678 T^{15} + 118546715477 T^{16} + 287162679051 T^{17} - 1042051568062 T^{18} - 18380602330 T^{19} + 22827114816044 T^{20} - 18380602330 p T^{21} - 1042051568062 p^{2} T^{22} + 287162679051 p^{3} T^{23} + 118546715477 p^{4} T^{24} - 20888923678 p^{5} T^{25} - 7858344544 p^{6} T^{26} + 1551192102 p^{7} T^{27} + 558528114 p^{8} T^{28} - 78317664 p^{9} T^{29} - 27101700 p^{10} T^{30} + 4946643 p^{11} T^{31} + 1355255 p^{12} T^{32} - 192056 p^{13} T^{33} - 48737 p^{14} T^{34} + 7647 p^{15} T^{35} + 90 p^{17} T^{36} - 222 p^{17} T^{37} - 36 p^{18} T^{38} + 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( ( 1 + 15 T + 192 T^{2} + 1589 T^{3} + 11979 T^{4} + 69512 T^{5} + 372551 T^{6} + 1588965 T^{7} + 6565755 T^{8} + 23376341 T^{9} + 107161144 T^{10} + 23376341 p T^{11} + 6565755 p^{2} T^{12} + 1588965 p^{3} T^{13} + 372551 p^{4} T^{14} + 69512 p^{5} T^{15} + 11979 p^{6} T^{16} + 1589 p^{7} T^{17} + 192 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( 1 + 14 T - 3 T^{2} - 674 T^{3} + 272 T^{4} + 32546 T^{5} + 81712 T^{6} - 666993 T^{7} - 3357348 T^{8} + 17322207 T^{9} + 184814474 T^{10} + 15819749 p T^{11} - 2165054635 T^{12} - 25568887291 T^{13} + 39755538703 T^{14} + 1356297700524 T^{15} + 3268702130064 T^{16} - 22207962657869 T^{17} - 130120648600979 T^{18} + 449732224613407 T^{19} + 6866741187502854 T^{20} + 449732224613407 p T^{21} - 130120648600979 p^{2} T^{22} - 22207962657869 p^{3} T^{23} + 3268702130064 p^{4} T^{24} + 1356297700524 p^{5} T^{25} + 39755538703 p^{6} T^{26} - 25568887291 p^{7} T^{27} - 2165054635 p^{8} T^{28} + 15819749 p^{10} T^{29} + 184814474 p^{10} T^{30} + 17322207 p^{11} T^{31} - 3357348 p^{12} T^{32} - 666993 p^{13} T^{33} + 81712 p^{14} T^{34} + 32546 p^{15} T^{35} + 272 p^{16} T^{36} - 674 p^{17} T^{37} - 3 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 11 T - 34 T^{2} + 896 T^{3} - 2255 T^{4} - 22392 T^{5} + 139932 T^{6} + 33068 T^{7} - 2341888 T^{8} - 677325 T^{9} + 98810293 T^{10} - 155739890 T^{11} - 5944528065 T^{12} + 38499841514 T^{13} + 52030546484 T^{14} - 1554682269428 T^{15} + 5850441051700 T^{16} + 17052052903016 T^{17} - 200366178686360 T^{18} + 57952245355402 T^{19} + 4304171952347175 T^{20} + 57952245355402 p T^{21} - 200366178686360 p^{2} T^{22} + 17052052903016 p^{3} T^{23} + 5850441051700 p^{4} T^{24} - 1554682269428 p^{5} T^{25} + 52030546484 p^{6} T^{26} + 38499841514 p^{7} T^{27} - 5944528065 p^{8} T^{28} - 155739890 p^{9} T^{29} + 98810293 p^{10} T^{30} - 677325 p^{11} T^{31} - 2341888 p^{12} T^{32} + 33068 p^{13} T^{33} + 139932 p^{14} T^{34} - 22392 p^{15} T^{35} - 2255 p^{16} T^{36} + 896 p^{17} T^{37} - 34 p^{18} T^{38} - 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 45 T + 982 T^{2} - 14218 T^{3} + 157205 T^{4} - 1424570 T^{5} + 10889943 T^{6} - 70652751 T^{7} + 380002380 T^{8} - 1492607096 T^{9} + 1362378435 T^{10} + 46408427840 T^{11} - 583806012927 T^{12} + 4712901825302 T^{13} - 29189123849351 T^{14} + 134963725100020 T^{15} - 324252211045654 T^{16} - 1899371916740023 T^{17} + 34953722551907716 T^{18} - 316465928130264311 T^{19} + 2154089418521731893 T^{20} - 316465928130264311 p T^{21} + 34953722551907716 p^{2} T^{22} - 1899371916740023 p^{3} T^{23} - 324252211045654 p^{4} T^{24} + 134963725100020 p^{5} T^{25} - 29189123849351 p^{6} T^{26} + 4712901825302 p^{7} T^{27} - 583806012927 p^{8} T^{28} + 46408427840 p^{9} T^{29} + 1362378435 p^{10} T^{30} - 1492607096 p^{11} T^{31} + 380002380 p^{12} T^{32} - 70652751 p^{13} T^{33} + 10889943 p^{14} T^{34} - 1424570 p^{15} T^{35} + 157205 p^{16} T^{36} - 14218 p^{17} T^{37} + 982 p^{18} T^{38} - 45 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 9 T - 56 T^{2} + 639 T^{3} - 707 T^{4} - 10463 T^{5} + 197029 T^{6} - 376997 T^{7} - 12892471 T^{8} + 47257038 T^{9} + 338745258 T^{10} - 2730070349 T^{11} + 11918748374 T^{12} + 41466740171 T^{13} - 1032925664833 T^{14} + 2265055622625 T^{15} + 21408341489190 T^{16} - 151712729261985 T^{17} + 618242322374010 T^{18} + 3205090844744586 T^{19} - 55618641547906837 T^{20} + 3205090844744586 p T^{21} + 618242322374010 p^{2} T^{22} - 151712729261985 p^{3} T^{23} + 21408341489190 p^{4} T^{24} + 2265055622625 p^{5} T^{25} - 1032925664833 p^{6} T^{26} + 41466740171 p^{7} T^{27} + 11918748374 p^{8} T^{28} - 2730070349 p^{9} T^{29} + 338745258 p^{10} T^{30} + 47257038 p^{11} T^{31} - 12892471 p^{12} T^{32} - 376997 p^{13} T^{33} + 197029 p^{14} T^{34} - 10463 p^{15} T^{35} - 707 p^{16} T^{36} + 639 p^{17} T^{37} - 56 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( ( 1 + 4 T + 227 T^{2} + 346 T^{3} + 24013 T^{4} - 15962 T^{5} + 1607112 T^{6} - 4822962 T^{7} + 79356741 T^{8} - 388237940 T^{9} + 3469981168 T^{10} - 388237940 p T^{11} + 79356741 p^{2} T^{12} - 4822962 p^{3} T^{13} + 1607112 p^{4} T^{14} - 15962 p^{5} T^{15} + 24013 p^{6} T^{16} + 346 p^{7} T^{17} + 227 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 47 | \( 1 - 39 T + 13 p T^{2} - 3887 T^{3} - 12653 T^{4} + 373736 T^{5} - 1506921 T^{6} - 17359053 T^{7} + 229125838 T^{8} - 661817919 T^{9} - 5685778060 T^{10} + 41441222668 T^{11} + 178697527373 T^{12} - 2983232197836 T^{13} + 5031872649936 T^{14} + 89836619054826 T^{15} - 161527944649695 T^{16} - 7093171569209056 T^{17} + 53609036568741653 T^{18} + 43448419606702574 T^{19} - 2142781167208010645 T^{20} + 43448419606702574 p T^{21} + 53609036568741653 p^{2} T^{22} - 7093171569209056 p^{3} T^{23} - 161527944649695 p^{4} T^{24} + 89836619054826 p^{5} T^{25} + 5031872649936 p^{6} T^{26} - 2983232197836 p^{7} T^{27} + 178697527373 p^{8} T^{28} + 41441222668 p^{9} T^{29} - 5685778060 p^{10} T^{30} - 661817919 p^{11} T^{31} + 229125838 p^{12} T^{32} - 17359053 p^{13} T^{33} - 1506921 p^{14} T^{34} + 373736 p^{15} T^{35} - 12653 p^{16} T^{36} - 3887 p^{17} T^{37} + 13 p^{19} T^{38} - 39 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 - 36 T + 495 T^{2} - 2058 T^{3} - 21927 T^{4} + 237600 T^{5} + 1026187 T^{6} - 27117181 T^{7} + 86230541 T^{8} + 1215046309 T^{9} - 10281085633 T^{10} - 13598938070 T^{11} + 395497451841 T^{12} + 317665130656 T^{13} - 14582798793729 T^{14} - 59244215952916 T^{15} + 898544145241732 T^{16} + 4678664361041223 T^{17} - 66156550556297728 T^{18} - 308965979607118201 T^{19} + 6613714218608591477 T^{20} - 308965979607118201 p T^{21} - 66156550556297728 p^{2} T^{22} + 4678664361041223 p^{3} T^{23} + 898544145241732 p^{4} T^{24} - 59244215952916 p^{5} T^{25} - 14582798793729 p^{6} T^{26} + 317665130656 p^{7} T^{27} + 395497451841 p^{8} T^{28} - 13598938070 p^{9} T^{29} - 10281085633 p^{10} T^{30} + 1215046309 p^{11} T^{31} + 86230541 p^{12} T^{32} - 27117181 p^{13} T^{33} + 1026187 p^{14} T^{34} + 237600 p^{15} T^{35} - 21927 p^{16} T^{36} - 2058 p^{17} T^{37} + 495 p^{18} T^{38} - 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 31 T + 331 T^{2} + 918 T^{3} + 141 T^{4} + 115940 T^{5} + 595623 T^{6} - 7685799 T^{7} + 15623954 T^{8} + 1187481227 T^{9} + 974134794 T^{10} - 60697810422 T^{11} + 224783056044 T^{12} + 4057521553171 T^{13} - 29214111231771 T^{14} - 258301852406265 T^{15} + 2043845782694201 T^{16} + 7504431726948343 T^{17} - 176844216848215818 T^{18} - 3055313007859600 p T^{19} + 11135061439626216256 T^{20} - 3055313007859600 p^{2} T^{21} - 176844216848215818 p^{2} T^{22} + 7504431726948343 p^{3} T^{23} + 2043845782694201 p^{4} T^{24} - 258301852406265 p^{5} T^{25} - 29214111231771 p^{6} T^{26} + 4057521553171 p^{7} T^{27} + 224783056044 p^{8} T^{28} - 60697810422 p^{9} T^{29} + 974134794 p^{10} T^{30} + 1187481227 p^{11} T^{31} + 15623954 p^{12} T^{32} - 7685799 p^{13} T^{33} + 595623 p^{14} T^{34} + 115940 p^{15} T^{35} + 141 p^{16} T^{36} + 918 p^{17} T^{37} + 331 p^{18} T^{38} + 31 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 7 T - 316 T^{2} - 3741 T^{3} + 36134 T^{4} + 815626 T^{5} + 74307 T^{6} - 97325390 T^{7} - 566371368 T^{8} + 6389990409 T^{9} + 86264886982 T^{10} - 73543432799 T^{11} - 7203817102318 T^{12} - 33186808848441 T^{13} + 332272119664900 T^{14} + 4100763285643652 T^{15} + 1027522867296534 T^{16} - 257743472833045600 T^{17} - 1565165844353659781 T^{18} + 6702877050374433373 T^{19} + \)\(13\!\cdots\!03\)\( T^{20} + 6702877050374433373 p T^{21} - 1565165844353659781 p^{2} T^{22} - 257743472833045600 p^{3} T^{23} + 1027522867296534 p^{4} T^{24} + 4100763285643652 p^{5} T^{25} + 332272119664900 p^{6} T^{26} - 33186808848441 p^{7} T^{27} - 7203817102318 p^{8} T^{28} - 73543432799 p^{9} T^{29} + 86264886982 p^{10} T^{30} + 6389990409 p^{11} T^{31} - 566371368 p^{12} T^{32} - 97325390 p^{13} T^{33} + 74307 p^{14} T^{34} + 815626 p^{15} T^{35} + 36134 p^{16} T^{36} - 3741 p^{17} T^{37} - 316 p^{18} T^{38} + 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( ( 1 - 13 T + 499 T^{2} - 6246 T^{3} + 123223 T^{4} - 1392005 T^{5} + 19456685 T^{6} - 191049336 T^{7} + 2135962805 T^{8} - 17945977232 T^{9} + 168249744910 T^{10} - 17945977232 p T^{11} + 2135962805 p^{2} T^{12} - 191049336 p^{3} T^{13} + 19456685 p^{4} T^{14} - 1392005 p^{5} T^{15} + 123223 p^{6} T^{16} - 6246 p^{7} T^{17} + 499 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 71 | \( 1 - 33 T + 480 T^{2} - 4454 T^{3} + 32283 T^{4} - 150612 T^{5} + 411245 T^{6} - 11231347 T^{7} + 242453173 T^{8} - 2971581409 T^{9} + 26659983428 T^{10} - 163205546397 T^{11} + 556233051805 T^{12} - 1806130257454 T^{13} + 44102807754164 T^{14} - 889290380530345 T^{15} + 10727938754070311 T^{16} - 86144470605875509 T^{17} + 467618479929279566 T^{18} - 1072810452954508994 T^{19} - 3270278006462267344 T^{20} - 1072810452954508994 p T^{21} + 467618479929279566 p^{2} T^{22} - 86144470605875509 p^{3} T^{23} + 10727938754070311 p^{4} T^{24} - 889290380530345 p^{5} T^{25} + 44102807754164 p^{6} T^{26} - 1806130257454 p^{7} T^{27} + 556233051805 p^{8} T^{28} - 163205546397 p^{9} T^{29} + 26659983428 p^{10} T^{30} - 2971581409 p^{11} T^{31} + 242453173 p^{12} T^{32} - 11231347 p^{13} T^{33} + 411245 p^{14} T^{34} - 150612 p^{15} T^{35} + 32283 p^{16} T^{36} - 4454 p^{17} T^{37} + 480 p^{18} T^{38} - 33 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 44 T + 966 T^{2} + 13918 T^{3} + 142165 T^{4} + 960784 T^{5} + 2028905 T^{6} - 42529828 T^{7} - 521222338 T^{8} - 324760926 T^{9} + 70442103629 T^{10} + 1235062443714 T^{11} + 12669262671426 T^{12} + 80505730855708 T^{13} + 140536666782370 T^{14} - 3458067691085104 T^{15} - 44106509738379217 T^{16} - 195579249667220486 T^{17} + 1763080522597000993 T^{18} + 44639395401991766024 T^{19} + \)\(48\!\cdots\!69\)\( T^{20} + 44639395401991766024 p T^{21} + 1763080522597000993 p^{2} T^{22} - 195579249667220486 p^{3} T^{23} - 44106509738379217 p^{4} T^{24} - 3458067691085104 p^{5} T^{25} + 140536666782370 p^{6} T^{26} + 80505730855708 p^{7} T^{27} + 12669262671426 p^{8} T^{28} + 1235062443714 p^{9} T^{29} + 70442103629 p^{10} T^{30} - 324760926 p^{11} T^{31} - 521222338 p^{12} T^{32} - 42529828 p^{13} T^{33} + 2028905 p^{14} T^{34} + 960784 p^{15} T^{35} + 142165 p^{16} T^{36} + 13918 p^{17} T^{37} + 966 p^{18} T^{38} + 44 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 21 T + 97 T^{2} + 1542 T^{3} - 23849 T^{4} - 15666 T^{5} + 2332680 T^{6} - 10507072 T^{7} - 119430209 T^{8} + 1497938247 T^{9} + 1461304924 T^{10} - 103008337298 T^{11} + 744923621846 T^{12} - 1743715779735 T^{13} - 61996476944778 T^{14} + 877728138558096 T^{15} - 2399910539901550 T^{16} - 90579048126065773 T^{17} + 919943151301647539 T^{18} + 2683704671247216766 T^{19} - 81935003669463238461 T^{20} + 2683704671247216766 p T^{21} + 919943151301647539 p^{2} T^{22} - 90579048126065773 p^{3} T^{23} - 2399910539901550 p^{4} T^{24} + 877728138558096 p^{5} T^{25} - 61996476944778 p^{6} T^{26} - 1743715779735 p^{7} T^{27} + 744923621846 p^{8} T^{28} - 103008337298 p^{9} T^{29} + 1461304924 p^{10} T^{30} + 1497938247 p^{11} T^{31} - 119430209 p^{12} T^{32} - 10507072 p^{13} T^{33} + 2332680 p^{14} T^{34} - 15666 p^{15} T^{35} - 23849 p^{16} T^{36} + 1542 p^{17} T^{37} + 97 p^{18} T^{38} - 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 25 T - 67 T^{2} - 6100 T^{3} - 14890 T^{4} + 665955 T^{5} + 822596 T^{6} - 57017500 T^{7} + 300210653 T^{8} + 5893483970 T^{9} - 47956998793 T^{10} - 446025792440 T^{11} + 4355496455818 T^{12} + 12458656511950 T^{13} - 368640371657529 T^{14} + 533730238559025 T^{15} + 23542624067679612 T^{16} - 33982757956830800 T^{17} - 350189686410790391 T^{18} + 327263258080523765 T^{19} - 47023156694501700397 T^{20} + 327263258080523765 p T^{21} - 350189686410790391 p^{2} T^{22} - 33982757956830800 p^{3} T^{23} + 23542624067679612 p^{4} T^{24} + 533730238559025 p^{5} T^{25} - 368640371657529 p^{6} T^{26} + 12458656511950 p^{7} T^{27} + 4355496455818 p^{8} T^{28} - 446025792440 p^{9} T^{29} - 47956998793 p^{10} T^{30} + 5893483970 p^{11} T^{31} + 300210653 p^{12} T^{32} - 57017500 p^{13} T^{33} + 822596 p^{14} T^{34} + 665955 p^{15} T^{35} - 14890 p^{16} T^{36} - 6100 p^{17} T^{37} - 67 p^{18} T^{38} + 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( ( 1 + T + 404 T^{2} + 1091 T^{3} + 91668 T^{4} + 296121 T^{5} + 14752237 T^{6} + 51759086 T^{7} + 1818815986 T^{8} + 6271509341 T^{9} + 180486356208 T^{10} + 6271509341 p T^{11} + 1818815986 p^{2} T^{12} + 51759086 p^{3} T^{13} + 14752237 p^{4} T^{14} + 296121 p^{5} T^{15} + 91668 p^{6} T^{16} + 1091 p^{7} T^{17} + 404 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 52 T + 1126 T^{2} - 11707 T^{3} + 13880 T^{4} + 1625978 T^{5} - 33273308 T^{6} + 384155551 T^{7} - 2101783118 T^{8} - 15658072569 T^{9} + 540703819819 T^{10} - 7443267945444 T^{11} + 60160143948102 T^{12} - 92418722175081 T^{13} - 5441467163519067 T^{14} + 97074194894803880 T^{15} - 972395796604666506 T^{16} + 5116273777308678085 T^{17} + 22151176958358021051 T^{18} - \)\(85\!\cdots\!67\)\( T^{19} + \)\(10\!\cdots\!45\)\( T^{20} - \)\(85\!\cdots\!67\)\( p T^{21} + 22151176958358021051 p^{2} T^{22} + 5116273777308678085 p^{3} T^{23} - 972395796604666506 p^{4} T^{24} + 97074194894803880 p^{5} T^{25} - 5441467163519067 p^{6} T^{26} - 92418722175081 p^{7} T^{27} + 60160143948102 p^{8} T^{28} - 7443267945444 p^{9} T^{29} + 540703819819 p^{10} T^{30} - 15658072569 p^{11} T^{31} - 2101783118 p^{12} T^{32} + 384155551 p^{13} T^{33} - 33273308 p^{14} T^{34} + 1625978 p^{15} T^{35} + 13880 p^{16} T^{36} - 11707 p^{17} T^{37} + 1126 p^{18} T^{38} - 52 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.45087030849985427802961478822, −2.34585896161526915751204412565, −2.24443524228328979019249320978, −2.19154514115572490328425865019, −2.16214118581339814072403443852, −2.10759658775407998160021818169, −2.05983102501156844014151452050, −2.04660887441372865000970872292, −2.02964203530198657635580677559, −1.97010222651520140735095025840, −1.80082171995197429323370283018, −1.63304599993737302842229316865, −1.59537997952688112322621652151, −1.55104856487725656861402360182, −1.35501911784903128243490981942, −1.32656504755212536751542160794, −1.18813447455133259331275385309, −0.959196513201054819678335032664, −0.922400146258374924568630681844, −0.910178924480856908817833419395, −0.808408506890329176293038780080, −0.75328801101032477600796312254, −0.57244728424920750480300603410, −0.53180331935903202936294219547, −0.28118741884041545759757373157, 0.28118741884041545759757373157, 0.53180331935903202936294219547, 0.57244728424920750480300603410, 0.75328801101032477600796312254, 0.808408506890329176293038780080, 0.910178924480856908817833419395, 0.922400146258374924568630681844, 0.959196513201054819678335032664, 1.18813447455133259331275385309, 1.32656504755212536751542160794, 1.35501911784903128243490981942, 1.55104856487725656861402360182, 1.59537997952688112322621652151, 1.63304599993737302842229316865, 1.80082171995197429323370283018, 1.97010222651520140735095025840, 2.02964203530198657635580677559, 2.04660887441372865000970872292, 2.05983102501156844014151452050, 2.10759658775407998160021818169, 2.16214118581339814072403443852, 2.19154514115572490328425865019, 2.24443524228328979019249320978, 2.34585896161526915751204412565, 2.45087030849985427802961478822
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.