Dirichlet series
| L(s) = 1 | + 2-s + 3·4-s + 2·5-s − 2·7-s + 2·8-s + 9-s + 2·10-s + 6·11-s − 6·13-s − 2·14-s + 4·16-s − 6·17-s + 18-s + 4·19-s + 6·20-s + 6·22-s + 6·25-s − 6·26-s − 6·28-s + 6·29-s − 6·34-s − 4·35-s + 3·36-s − 2·37-s + 4·38-s + 4·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s + 0.707·8-s + 1/3·9-s + 0.632·10-s + 1.80·11-s − 1.66·13-s − 0.534·14-s + 16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 1.34·20-s + 1.27·22-s + 6/5·25-s − 1.17·26-s − 1.13·28-s + 1.11·29-s − 1.02·34-s − 0.676·35-s + 1/2·36-s − 0.328·37-s + 0.648·38-s + 0.632·40-s − 0.312·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(23^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.27081\times 10^{12}\) |
| Root analytic conductor: | \(2.05525\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 23^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(14.54281124\) |
| \(L(\frac12)\) | \(\approx\) | \(14.54281124\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - T - p T^{2} + 3 T^{3} + T^{4} - p T^{5} + T^{6} - 9 T^{7} + 3 p T^{8} + 27 T^{9} - 31 T^{10} - 3 T^{11} + 9 p T^{12} - 55 T^{13} + 131 T^{14} + 5 p T^{15} - 365 T^{16} + 293 T^{17} + 129 p T^{18} - 447 T^{19} + 547 T^{20} - 447 p T^{21} + 129 p^{3} T^{22} + 293 p^{3} T^{23} - 365 p^{4} T^{24} + 5 p^{6} T^{25} + 131 p^{6} T^{26} - 55 p^{7} T^{27} + 9 p^{9} T^{28} - 3 p^{9} T^{29} - 31 p^{10} T^{30} + 27 p^{11} T^{31} + 3 p^{13} T^{32} - 9 p^{13} T^{33} + p^{14} T^{34} - p^{16} T^{35} + p^{16} T^{36} + 3 p^{17} T^{37} - p^{19} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) |
| 3 | \( 1 - T^{2} - 8 T^{4} + 17 T^{6} + 55 T^{8} - 208 T^{10} - 287 T^{12} + 2159 T^{14} + 424 T^{16} - 19855 T^{18} + 16039 T^{20} - 19855 p^{2} T^{22} + 424 p^{4} T^{24} + 2159 p^{6} T^{26} - 287 p^{8} T^{28} - 208 p^{10} T^{30} + 55 p^{12} T^{32} + 17 p^{14} T^{34} - 8 p^{16} T^{36} - p^{18} T^{38} + p^{20} T^{40} \) | |
| 5 | \( 1 - 2 T - 2 T^{2} + 6 T^{3} - p T^{4} + 44 T^{5} - 68 T^{6} - 228 T^{7} + 549 T^{8} - 6 p^{2} T^{9} + 986 T^{10} - 1404 p T^{11} - 4101 T^{12} + 44212 T^{13} - 18268 T^{14} - 12226 T^{15} - 41107 p T^{16} - 438194 T^{17} + 2688558 T^{18} - 386952 T^{19} - 5837129 T^{20} - 386952 p T^{21} + 2688558 p^{2} T^{22} - 438194 p^{3} T^{23} - 41107 p^{5} T^{24} - 12226 p^{5} T^{25} - 18268 p^{6} T^{26} + 44212 p^{7} T^{27} - 4101 p^{8} T^{28} - 1404 p^{10} T^{29} + 986 p^{10} T^{30} - 6 p^{13} T^{31} + 549 p^{12} T^{32} - 228 p^{13} T^{33} - 68 p^{14} T^{34} + 44 p^{15} T^{35} - p^{17} T^{36} + 6 p^{17} T^{37} - 2 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 + 2 T - 6 T^{2} - 18 T^{3} + 3 T^{4} + 4 T^{5} + 20 T^{6} + 132 p T^{7} + 1557 T^{8} - 6042 T^{9} - 15698 T^{10} + 31812 T^{11} + 59723 T^{12} - 17484 p T^{13} + 372564 T^{14} + 1246342 T^{15} - 5872815 T^{16} - 12996142 T^{17} + 31929018 T^{18} + 50529288 T^{19} - 112409657 T^{20} + 50529288 p T^{21} + 31929018 p^{2} T^{22} - 12996142 p^{3} T^{23} - 5872815 p^{4} T^{24} + 1246342 p^{5} T^{25} + 372564 p^{6} T^{26} - 17484 p^{8} T^{27} + 59723 p^{8} T^{28} + 31812 p^{9} T^{29} - 15698 p^{10} T^{30} - 6042 p^{11} T^{31} + 1557 p^{12} T^{32} + 132 p^{14} T^{33} + 20 p^{14} T^{34} + 4 p^{15} T^{35} + 3 p^{16} T^{36} - 18 p^{17} T^{37} - 6 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 6 T + 10 T^{2} + 30 T^{3} - 15 p T^{4} + 276 T^{5} - 556 T^{6} + 3420 T^{7} - 4315 T^{8} - 5430 p T^{9} + 312126 T^{10} - 80100 p T^{11} + 1635619 T^{12} - 278100 T^{13} - 20472140 T^{14} + 128725686 T^{15} - 38148525 p T^{16} + 555746154 T^{17} + 1557234890 T^{18} - 11672788200 T^{19} + 43645062671 T^{20} - 11672788200 p T^{21} + 1557234890 p^{2} T^{22} + 555746154 p^{3} T^{23} - 38148525 p^{5} T^{24} + 128725686 p^{5} T^{25} - 20472140 p^{6} T^{26} - 278100 p^{7} T^{27} + 1635619 p^{8} T^{28} - 80100 p^{10} T^{29} + 312126 p^{10} T^{30} - 5430 p^{12} T^{31} - 4315 p^{12} T^{32} + 3420 p^{13} T^{33} - 556 p^{14} T^{34} + 276 p^{15} T^{35} - 15 p^{17} T^{36} + 30 p^{17} T^{37} + 10 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( ( 1 + 3 T - 4 T^{2} - 51 T^{3} - 101 T^{4} + 360 T^{5} + 2393 T^{6} + 2499 T^{7} - 23612 T^{8} - 103323 T^{9} - 3013 T^{10} - 103323 p T^{11} - 23612 p^{2} T^{12} + 2499 p^{3} T^{13} + 2393 p^{4} T^{14} + 360 p^{5} T^{15} - 101 p^{6} T^{16} - 51 p^{7} T^{17} - 4 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 17 | \( 1 + 6 T - 2 T^{2} - 138 T^{3} - 429 T^{4} + 732 T^{5} + 7196 T^{6} + 11484 T^{7} - 347 p T^{8} + 50658 T^{9} - 380166 T^{10} + 13085028 T^{11} + 345427 p^{2} T^{12} + 48322260 T^{13} - 2059004924 T^{14} - 7789351410 T^{15} + 7584540705 T^{16} + 117518962422 T^{17} + 217439806958 T^{18} - 136336020888 T^{19} + 714273113567 T^{20} - 136336020888 p T^{21} + 217439806958 p^{2} T^{22} + 117518962422 p^{3} T^{23} + 7584540705 p^{4} T^{24} - 7789351410 p^{5} T^{25} - 2059004924 p^{6} T^{26} + 48322260 p^{7} T^{27} + 345427 p^{10} T^{28} + 13085028 p^{9} T^{29} - 380166 p^{10} T^{30} + 50658 p^{11} T^{31} - 347 p^{13} T^{32} + 11484 p^{13} T^{33} + 7196 p^{14} T^{34} + 732 p^{15} T^{35} - 429 p^{16} T^{36} - 138 p^{17} T^{37} - 2 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( ( 1 - 2 T - 15 T^{2} + 68 T^{3} + 149 T^{4} - 1590 T^{5} + 349 T^{6} + 29512 T^{7} - 65655 T^{8} - 429418 T^{9} + 2106281 T^{10} - 429418 p T^{11} - 65655 p^{2} T^{12} + 29512 p^{3} T^{13} + 349 p^{4} T^{14} - 1590 p^{5} T^{15} + 149 p^{6} T^{16} + 68 p^{7} T^{17} - 15 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 3 T - 20 T^{2} + 147 T^{3} + 139 T^{4} - 4680 T^{5} + 10009 T^{6} + 105693 T^{7} - 607340 T^{8} - 1243077 T^{9} + 21342091 T^{10} - 1243077 p T^{11} - 607340 p^{2} T^{12} + 105693 p^{3} T^{13} + 10009 p^{4} T^{14} - 4680 p^{5} T^{15} + 139 p^{6} T^{16} + 147 p^{7} T^{17} - 20 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( 1 - 17 T^{2} - 672 T^{4} + 27761 T^{6} + 173855 T^{8} - 29633856 T^{10} + 336700897 T^{12} + 22754220367 T^{14} - 710391308256 T^{16} - 9790153532335 T^{18} + 849118657283711 T^{20} - 9790153532335 p^{2} T^{22} - 710391308256 p^{4} T^{24} + 22754220367 p^{6} T^{26} + 336700897 p^{8} T^{28} - 29633856 p^{10} T^{30} + 173855 p^{12} T^{32} + 27761 p^{14} T^{34} - 672 p^{16} T^{36} - 17 p^{18} T^{38} + p^{20} T^{40} \) | |
| 37 | \( 1 + 2 T - 66 T^{2} - 198 T^{3} + 3003 T^{4} + 12244 T^{5} - 110020 T^{6} - 583836 T^{7} + 3328677 T^{8} + 22622358 T^{9} - 80349158 T^{10} - 1306893588 T^{11} + 127749563 T^{12} + 54822204492 T^{13} + 113989811364 T^{14} - 1810983902078 T^{15} - 7719300618975 T^{16} + 44713919998898 T^{17} + 339714022814958 T^{18} - 542537638152312 T^{19} - 10988504246056457 T^{20} - 542537638152312 p T^{21} + 339714022814958 p^{2} T^{22} + 44713919998898 p^{3} T^{23} - 7719300618975 p^{4} T^{24} - 1810983902078 p^{5} T^{25} + 113989811364 p^{6} T^{26} + 54822204492 p^{7} T^{27} + 127749563 p^{8} T^{28} - 1306893588 p^{9} T^{29} - 80349158 p^{10} T^{30} + 22622358 p^{11} T^{31} + 3328677 p^{12} T^{32} - 583836 p^{13} T^{33} - 110020 p^{14} T^{34} + 12244 p^{15} T^{35} + 3003 p^{16} T^{36} - 198 p^{17} T^{37} - 66 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 2 T - 59 T^{2} - 162 T^{3} + 1876 T^{4} + 5758 T^{5} - 20777 T^{6} + 21846 T^{7} - 1328757 T^{8} - 15416724 T^{9} + 89595752 T^{10} - 643168086 T^{11} - 5961399399 T^{12} + 61859155328 T^{13} + 295936230629 T^{14} - 2712923522558 T^{15} - 8976266448128 T^{16} + 73243179830108 T^{17} - 7937386644825 T^{18} - 805824509912952 T^{19} + 19983450984170095 T^{20} - 805824509912952 p T^{21} - 7937386644825 p^{2} T^{22} + 73243179830108 p^{3} T^{23} - 8976266448128 p^{4} T^{24} - 2712923522558 p^{5} T^{25} + 295936230629 p^{6} T^{26} + 61859155328 p^{7} T^{27} - 5961399399 p^{8} T^{28} - 643168086 p^{9} T^{29} + 89595752 p^{10} T^{30} - 15416724 p^{11} T^{31} - 1328757 p^{12} T^{32} + 21846 p^{13} T^{33} - 20777 p^{14} T^{34} + 5758 p^{15} T^{35} + 1876 p^{16} T^{36} - 162 p^{17} T^{37} - 59 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} - p^{9} T^{18} + p^{10} T^{20} )^{2} \) | |
| 47 | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{10} \) | |
| 53 | \( 1 - 8 T - 38 T^{2} + 696 T^{3} - 1109 T^{4} - 23536 T^{5} + 113044 T^{6} + 11472 T^{7} + 27777 p T^{8} + 5234376 T^{9} - 514442194 T^{10} + 1878990384 T^{11} + 586536771 p T^{12} - 236928482000 T^{13} - 627013727896 T^{14} + 10724076680600 T^{15} - 8701371687575 T^{16} - 92858921354696 T^{17} - 1155315667944318 T^{18} - 7720414478540064 T^{19} + 243419849683430227 T^{20} - 7720414478540064 p T^{21} - 1155315667944318 p^{2} T^{22} - 92858921354696 p^{3} T^{23} - 8701371687575 p^{4} T^{24} + 10724076680600 p^{5} T^{25} - 627013727896 p^{6} T^{26} - 236928482000 p^{7} T^{27} + 586536771 p^{9} T^{28} + 1878990384 p^{9} T^{29} - 514442194 p^{10} T^{30} + 5234376 p^{11} T^{31} + 27777 p^{13} T^{32} + 11472 p^{13} T^{33} + 113044 p^{14} T^{34} - 23536 p^{15} T^{35} - 1109 p^{16} T^{36} + 696 p^{17} T^{37} - 38 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 4 T - 86 T^{2} - 516 T^{3} + 4171 T^{4} + 35096 T^{5} - 107468 T^{6} - 1229112 T^{7} - 191307 T^{8} - 22927428 T^{9} + 11829278 T^{10} - 15083917368 T^{11} - 66287189169 T^{12} + 1356012901336 T^{13} + 8584362685016 T^{14} - 67112525482084 T^{15} - 593290345587383 T^{16} + 1777944900936196 T^{17} + 21906672333345330 T^{18} - 121510917598656 T^{19} + 249872067939004195 T^{20} - 121510917598656 p T^{21} + 21906672333345330 p^{2} T^{22} + 1777944900936196 p^{3} T^{23} - 593290345587383 p^{4} T^{24} - 67112525482084 p^{5} T^{25} + 8584362685016 p^{6} T^{26} + 1356012901336 p^{7} T^{27} - 66287189169 p^{8} T^{28} - 15083917368 p^{9} T^{29} + 11829278 p^{10} T^{30} - 22927428 p^{11} T^{31} - 191307 p^{12} T^{32} - 1229112 p^{13} T^{33} - 107468 p^{14} T^{34} + 35096 p^{15} T^{35} + 4171 p^{16} T^{36} - 516 p^{17} T^{37} - 86 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 4 T - 30 T^{2} - 60 T^{3} - 1605 T^{4} - 424 p T^{5} + 67364 T^{6} + 1290840 T^{7} + 1726005 T^{8} + 60202140 T^{9} + 225715846 T^{10} + 16336654920 T^{11} + 63230548319 T^{12} - 667501429560 T^{13} - 2432355803400 T^{14} - 399750764284 p T^{15} - 383802038636535 T^{16} + 1476770493397924 T^{17} + 26662881319720650 T^{18} + 35808213383210160 T^{19} + 705199698981330931 T^{20} + 35808213383210160 p T^{21} + 26662881319720650 p^{2} T^{22} + 1476770493397924 p^{3} T^{23} - 383802038636535 p^{4} T^{24} - 399750764284 p^{6} T^{25} - 2432355803400 p^{6} T^{26} - 667501429560 p^{7} T^{27} + 63230548319 p^{8} T^{28} + 16336654920 p^{9} T^{29} + 225715846 p^{10} T^{30} + 60202140 p^{11} T^{31} + 1726005 p^{12} T^{32} + 1290840 p^{13} T^{33} + 67364 p^{14} T^{34} - 424 p^{16} T^{35} - 1605 p^{16} T^{36} - 60 p^{17} T^{37} - 30 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 10 T - 54 T^{2} + 1410 T^{3} - 3573 T^{4} - 100340 T^{5} + 851348 T^{6} + 3003300 T^{7} - 77873595 T^{8} + 196250850 T^{9} + 4196112958 T^{10} - 33682545300 T^{11} - 91289444077 T^{12} + 2407640669460 T^{13} - 6286878024204 T^{14} - 95543009471510 T^{15} + 720288376365969 T^{16} + 915049005963590 T^{17} - 27839288223431670 T^{18} + 83774692668333000 T^{19} - 396953895777292961 T^{20} + 83774692668333000 p T^{21} - 27839288223431670 p^{2} T^{22} + 915049005963590 p^{3} T^{23} + 720288376365969 p^{4} T^{24} - 95543009471510 p^{5} T^{25} - 6286878024204 p^{6} T^{26} + 2407640669460 p^{7} T^{27} - 91289444077 p^{8} T^{28} - 33682545300 p^{9} T^{29} + 4196112958 p^{10} T^{30} + 196250850 p^{11} T^{31} - 77873595 p^{12} T^{32} + 3003300 p^{13} T^{33} + 851348 p^{14} T^{34} - 100340 p^{15} T^{35} - 3573 p^{16} T^{36} + 1410 p^{17} T^{37} - 54 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 20 T + 163 T^{2} - 60 T^{3} - 232 p T^{4} - 184580 T^{5} - 694619 T^{6} + 6765300 T^{7} + 120862455 T^{8} + 10675800 p T^{9} - 376465456 T^{10} - 863324700 p T^{11} - 669632240583 T^{12} - 3221047286080 T^{13} + 9139261406267 T^{14} + 304289060000140 T^{15} + 38331059510744 p T^{16} + 11528647851251720 T^{17} - 28404332911048035 T^{18} - 969759930555114000 T^{19} - 10011799602956349089 T^{20} - 969759930555114000 p T^{21} - 28404332911048035 p^{2} T^{22} + 11528647851251720 p^{3} T^{23} + 38331059510744 p^{5} T^{24} + 304289060000140 p^{5} T^{25} + 9139261406267 p^{6} T^{26} - 3221047286080 p^{7} T^{27} - 669632240583 p^{8} T^{28} - 863324700 p^{10} T^{29} - 376465456 p^{10} T^{30} + 10675800 p^{12} T^{31} + 120862455 p^{12} T^{32} + 6765300 p^{13} T^{33} - 694619 p^{14} T^{34} - 184580 p^{15} T^{35} - 232 p^{17} T^{36} - 60 p^{17} T^{37} + 163 p^{18} T^{38} + 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 22 T + 237 T^{2} + 1386 T^{3} + 1956 T^{4} - 35926 T^{5} - 310561 T^{6} - 2203278 T^{7} - 39884229 T^{8} - 640554684 T^{9} - 6124283384 T^{10} - 65332668786 T^{11} - 740808483607 T^{12} - 6582700652880 T^{13} - 34127678829411 T^{14} + 33537494302070 T^{15} + 2543312706126480 T^{16} + 27939278021334724 T^{17} + 222193494387179007 T^{18} + 1893094104151657176 T^{19} + 18083444269018797967 T^{20} + 1893094104151657176 p T^{21} + 222193494387179007 p^{2} T^{22} + 27939278021334724 p^{3} T^{23} + 2543312706126480 p^{4} T^{24} + 33537494302070 p^{5} T^{25} - 34127678829411 p^{6} T^{26} - 6582700652880 p^{7} T^{27} - 740808483607 p^{8} T^{28} - 65332668786 p^{9} T^{29} - 6124283384 p^{10} T^{30} - 640554684 p^{11} T^{31} - 39884229 p^{12} T^{32} - 2203278 p^{13} T^{33} - 310561 p^{14} T^{34} - 35926 p^{15} T^{35} + 1956 p^{16} T^{36} + 1386 p^{17} T^{37} + 237 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 4 T - 66 T^{2} + 276 T^{3} - 669 T^{4} + 25864 T^{5} + 276092 T^{6} - 4736328 T^{7} - 7692027 T^{8} + 170484708 T^{9} - 277603142 T^{10} - 79559884152 T^{11} + 335135767031 T^{12} + 4207094962584 T^{13} - 17436068145624 T^{14} + 115392820261444 T^{15} - 2452838023321623 T^{16} - 20397541295621476 T^{17} + 355077253189082070 T^{18} + 307219597602593616 T^{19} - 8591427998948468645 T^{20} + 307219597602593616 p T^{21} + 355077253189082070 p^{2} T^{22} - 20397541295621476 p^{3} T^{23} - 2452838023321623 p^{4} T^{24} + 115392820261444 p^{5} T^{25} - 17436068145624 p^{6} T^{26} + 4207094962584 p^{7} T^{27} + 335135767031 p^{8} T^{28} - 79559884152 p^{9} T^{29} - 277603142 p^{10} T^{30} + 170484708 p^{11} T^{31} - 7692027 p^{12} T^{32} - 4736328 p^{13} T^{33} + 276092 p^{14} T^{34} + 25864 p^{15} T^{35} - 669 p^{16} T^{36} + 276 p^{17} T^{37} - 66 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 22 T + 202 T^{2} - 66 T^{3} - 22229 T^{4} + 290356 T^{5} - 1390316 T^{6} - 10248612 T^{7} + 240484101 T^{8} - 1862087106 T^{9} + 1441252286 T^{10} + 141331308756 T^{11} - 1772245853517 T^{12} + 9330171107180 T^{13} + 26509809547124 T^{14} - 951834519759530 T^{15} + 8636702385588625 T^{16} - 34272578901173734 T^{17} - 126129581799405558 T^{18} + 3226287500256488184 T^{19} - 32326296598987172033 T^{20} + 3226287500256488184 p T^{21} - 126129581799405558 p^{2} T^{22} - 34272578901173734 p^{3} T^{23} + 8636702385588625 p^{4} T^{24} - 951834519759530 p^{5} T^{25} + 26509809547124 p^{6} T^{26} + 9330171107180 p^{7} T^{27} - 1772245853517 p^{8} T^{28} + 141331308756 p^{9} T^{29} + 1441252286 p^{10} T^{30} - 1862087106 p^{11} T^{31} + 240484101 p^{12} T^{32} - 10248612 p^{13} T^{33} - 1390316 p^{14} T^{34} + 290356 p^{15} T^{35} - 22229 p^{16} T^{36} - 66 p^{17} T^{37} + 202 p^{18} T^{38} - 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 12 T - 50 T^{2} + 1860 T^{3} - 7725 T^{4} - 119688 T^{5} + 1344476 T^{6} + 603000 T^{7} - 79047835 T^{8} + 343740300 T^{9} - 934854 p T^{10} - 87831468360 T^{11} + 1329142070119 T^{12} - 1544307153480 T^{13} - 144858824463320 T^{14} + 1314090811327428 T^{15} + 3454707912458985 T^{16} - 129448430857488012 T^{17} + 627145597349077670 T^{18} + 3488707051132989120 T^{19} - 52644547718107360789 T^{20} + 3488707051132989120 p T^{21} + 627145597349077670 p^{2} T^{22} - 129448430857488012 p^{3} T^{23} + 3454707912458985 p^{4} T^{24} + 1314090811327428 p^{5} T^{25} - 144858824463320 p^{6} T^{26} - 1544307153480 p^{7} T^{27} + 1329142070119 p^{8} T^{28} - 87831468360 p^{9} T^{29} - 934854 p^{11} T^{30} + 343740300 p^{11} T^{31} - 79047835 p^{12} T^{32} + 603000 p^{13} T^{33} + 1344476 p^{14} T^{34} - 119688 p^{15} T^{35} - 7725 p^{16} T^{36} + 1860 p^{17} T^{37} - 50 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 22 T + 214 T^{2} + 902 T^{3} - 397 T^{4} - 2596 T^{5} - 38580 T^{6} - 9481956 T^{7} - 199990923 T^{8} - 1897576142 T^{9} - 7620620798 T^{10} + 73568529012 T^{11} + 1508362361203 T^{12} + 14912358250372 T^{13} + 149511705983364 T^{14} + 1789559793366862 T^{15} + 16632945888297505 T^{16} + 61491267124240998 T^{17} - 725922555659612762 T^{18} - 13916199918218479112 T^{19} - \)\(13\!\cdots\!37\)\( T^{20} - 13916199918218479112 p T^{21} - 725922555659612762 p^{2} T^{22} + 61491267124240998 p^{3} T^{23} + 16632945888297505 p^{4} T^{24} + 1789559793366862 p^{5} T^{25} + 149511705983364 p^{6} T^{26} + 14912358250372 p^{7} T^{27} + 1508362361203 p^{8} T^{28} + 73568529012 p^{9} T^{29} - 7620620798 p^{10} T^{30} - 1897576142 p^{11} T^{31} - 199990923 p^{12} T^{32} - 9481956 p^{13} T^{33} - 38580 p^{14} T^{34} - 2596 p^{15} T^{35} - 397 p^{16} T^{36} + 902 p^{17} T^{37} + 214 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.55251207550192598626392230033, −2.50029619237878292672135820700, −2.41454698462316388713017298989, −2.40122388005194302294267150539, −2.39811987723132095052995566772, −2.31453573086919654181379519262, −2.07318560345634126918617052889, −1.95934744516097019506284665102, −1.93281287886989059711812707413, −1.93064368649967139945652901033, −1.78053194592699271368031723161, −1.73249023582624203586165813399, −1.45839984849032635853241413954, −1.43665508999697711885941810589, −1.41967484067096512604098432793, −1.33471246985692708175489131292, −1.32409595106838190143340645655, −1.28034291387521534481272424006, −1.23553966420902331056377129713, −0.871144733742933195625084165583, −0.797284264170690173304066381933, −0.76195252838947469036014382491, −0.39509115621935427544051490318, −0.32970530904491002570103110089, −0.20590961816314031028362261275, 0.20590961816314031028362261275, 0.32970530904491002570103110089, 0.39509115621935427544051490318, 0.76195252838947469036014382491, 0.797284264170690173304066381933, 0.871144733742933195625084165583, 1.23553966420902331056377129713, 1.28034291387521534481272424006, 1.32409595106838190143340645655, 1.33471246985692708175489131292, 1.41967484067096512604098432793, 1.43665508999697711885941810589, 1.45839984849032635853241413954, 1.73249023582624203586165813399, 1.78053194592699271368031723161, 1.93064368649967139945652901033, 1.93281287886989059711812707413, 1.95934744516097019506284665102, 2.07318560345634126918617052889, 2.31453573086919654181379519262, 2.39811987723132095052995566772, 2.40122388005194302294267150539, 2.41454698462316388713017298989, 2.50029619237878292672135820700, 2.55251207550192598626392230033
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.