Dirichlet series
| L(s) = 1 | − 6·2-s − 7·3-s + 9·4-s − 21·5-s + 42·6-s − 11·7-s + 18·8-s + 9·9-s + 126·10-s − 7·11-s − 63·12-s − 16·13-s + 66·14-s + 147·15-s − 69·16-s + 16·17-s − 54·18-s + 19-s − 189·20-s + 77·21-s + 42·22-s − 8·23-s − 126·24-s + 192·25-s + 96·26-s + 56·27-s − 99·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 4.04·3-s + 9/2·4-s − 9.39·5-s + 17.1·6-s − 4.15·7-s + 6.36·8-s + 3·9-s + 39.8·10-s − 2.11·11-s − 18.1·12-s − 4.43·13-s + 17.6·14-s + 37.9·15-s − 17.2·16-s + 3.88·17-s − 12.7·18-s + 0.229·19-s − 42.2·20-s + 16.8·21-s + 8.95·22-s − 1.66·23-s − 25.7·24-s + 38.3·25-s + 18.8·26-s + 10.7·27-s − 18.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(17^{16} \cdot 59^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.86581\times 10^{14}\) |
| Root analytic conductor: | \(2.83001\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 17^{16} \cdot 59^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( ( 1 - T )^{16} \) |
| 59 | \( ( 1 - T )^{16} \) | |
| good | 2 | \( 1 + 3 p T + 27 T^{2} + 45 p T^{3} + 129 p T^{4} + 159 p^{2} T^{5} + 1415 T^{6} + 2841 T^{7} + 5267 T^{8} + 9017 T^{9} + 14477 T^{10} + 21833 T^{11} + 31441 T^{12} + 43557 T^{13} + 59447 T^{14} + 20273 p^{2} T^{15} + 113283 T^{16} + 20273 p^{3} T^{17} + 59447 p^{2} T^{18} + 43557 p^{3} T^{19} + 31441 p^{4} T^{20} + 21833 p^{5} T^{21} + 14477 p^{6} T^{22} + 9017 p^{7} T^{23} + 5267 p^{8} T^{24} + 2841 p^{9} T^{25} + 1415 p^{10} T^{26} + 159 p^{13} T^{27} + 129 p^{13} T^{28} + 45 p^{14} T^{29} + 27 p^{14} T^{30} + 3 p^{16} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + 7 T + 40 T^{2} + 161 T^{3} + 587 T^{4} + 1837 T^{5} + 5398 T^{6} + 14467 T^{7} + 4091 p^{2} T^{8} + 9742 p^{2} T^{9} + 199691 T^{10} + 431612 T^{11} + 298832 p T^{12} + 1779482 T^{13} + 378010 p^{2} T^{14} + 692878 p^{2} T^{15} + 408304 p^{3} T^{16} + 692878 p^{3} T^{17} + 378010 p^{4} T^{18} + 1779482 p^{3} T^{19} + 298832 p^{5} T^{20} + 431612 p^{5} T^{21} + 199691 p^{6} T^{22} + 9742 p^{9} T^{23} + 4091 p^{10} T^{24} + 14467 p^{9} T^{25} + 5398 p^{10} T^{26} + 1837 p^{11} T^{27} + 587 p^{12} T^{28} + 161 p^{13} T^{29} + 40 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 + 21 T + 249 T^{2} + 2143 T^{3} + 14807 T^{4} + 17307 p T^{5} + 88321 p T^{6} + 2009432 T^{7} + 8272917 T^{8} + 31143873 T^{9} + 108048711 T^{10} + 347499761 T^{11} + 1040653133 T^{12} + 2911461051 T^{13} + 1525603264 p T^{14} + 18746200712 T^{15} + 43253965373 T^{16} + 18746200712 p T^{17} + 1525603264 p^{3} T^{18} + 2911461051 p^{3} T^{19} + 1040653133 p^{4} T^{20} + 347499761 p^{5} T^{21} + 108048711 p^{6} T^{22} + 31143873 p^{7} T^{23} + 8272917 p^{8} T^{24} + 2009432 p^{9} T^{25} + 88321 p^{11} T^{26} + 17307 p^{12} T^{27} + 14807 p^{12} T^{28} + 2143 p^{13} T^{29} + 249 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 11 T + 99 T^{2} + 631 T^{3} + 3529 T^{4} + 16644 T^{5} + 10230 p T^{6} + 39362 p T^{7} + 992543 T^{8} + 3302021 T^{9} + 10512336 T^{10} + 31647691 T^{11} + 92926579 T^{12} + 262444709 T^{13} + 731975367 T^{14} + 1980226443 T^{15} + 5314535269 T^{16} + 1980226443 p T^{17} + 731975367 p^{2} T^{18} + 262444709 p^{3} T^{19} + 92926579 p^{4} T^{20} + 31647691 p^{5} T^{21} + 10512336 p^{6} T^{22} + 3302021 p^{7} T^{23} + 992543 p^{8} T^{24} + 39362 p^{10} T^{25} + 10230 p^{11} T^{26} + 16644 p^{11} T^{27} + 3529 p^{12} T^{28} + 631 p^{13} T^{29} + 99 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 7 T + 9 p T^{2} + 573 T^{3} + 4855 T^{4} + 24373 T^{5} + 158595 T^{6} + 708606 T^{7} + 3877077 T^{8} + 15687965 T^{9} + 75519819 T^{10} + 280177541 T^{11} + 1218630412 T^{12} + 4181500767 T^{13} + 16705740691 T^{14} + 53294814856 T^{15} + 197433330166 T^{16} + 53294814856 p T^{17} + 16705740691 p^{2} T^{18} + 4181500767 p^{3} T^{19} + 1218630412 p^{4} T^{20} + 280177541 p^{5} T^{21} + 75519819 p^{6} T^{22} + 15687965 p^{7} T^{23} + 3877077 p^{8} T^{24} + 708606 p^{9} T^{25} + 158595 p^{10} T^{26} + 24373 p^{11} T^{27} + 4855 p^{12} T^{28} + 573 p^{13} T^{29} + 9 p^{15} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 16 T + 203 T^{2} + 1792 T^{3} + 14275 T^{4} + 95415 T^{5} + 600564 T^{6} + 3371571 T^{7} + 18169721 T^{8} + 89901243 T^{9} + 431538912 T^{10} + 1930538895 T^{11} + 8424397174 T^{12} + 2656927202 p T^{13} + 138565115677 T^{14} + 524517883170 T^{15} + 1945586553346 T^{16} + 524517883170 p T^{17} + 138565115677 p^{2} T^{18} + 2656927202 p^{4} T^{19} + 8424397174 p^{4} T^{20} + 1930538895 p^{5} T^{21} + 431538912 p^{6} T^{22} + 89901243 p^{7} T^{23} + 18169721 p^{8} T^{24} + 3371571 p^{9} T^{25} + 600564 p^{10} T^{26} + 95415 p^{11} T^{27} + 14275 p^{12} T^{28} + 1792 p^{13} T^{29} + 203 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - T + 107 T^{2} - 107 T^{3} + 6348 T^{4} - 4850 T^{5} + 273578 T^{6} - 134840 T^{7} + 9529325 T^{8} - 2760589 T^{9} + 282243356 T^{10} - 46032983 T^{11} + 7281865981 T^{12} - 636489901 T^{13} + 165561919843 T^{14} - 8211582005 T^{15} + 3336711850579 T^{16} - 8211582005 p T^{17} + 165561919843 p^{2} T^{18} - 636489901 p^{3} T^{19} + 7281865981 p^{4} T^{20} - 46032983 p^{5} T^{21} + 282243356 p^{6} T^{22} - 2760589 p^{7} T^{23} + 9529325 p^{8} T^{24} - 134840 p^{9} T^{25} + 273578 p^{10} T^{26} - 4850 p^{11} T^{27} + 6348 p^{12} T^{28} - 107 p^{13} T^{29} + 107 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 8 T + 160 T^{2} + 1156 T^{3} + 13275 T^{4} + 89694 T^{5} + 775926 T^{6} + 4914858 T^{7} + 1556178 p T^{8} + 211598797 T^{9} + 1371736394 T^{10} + 7558519108 T^{11} + 44803889428 T^{12} + 10023439848 p T^{13} + 1264759463636 T^{14} + 6088439654145 T^{15} + 31112640363596 T^{16} + 6088439654145 p T^{17} + 1264759463636 p^{2} T^{18} + 10023439848 p^{4} T^{19} + 44803889428 p^{4} T^{20} + 7558519108 p^{5} T^{21} + 1371736394 p^{6} T^{22} + 211598797 p^{7} T^{23} + 1556178 p^{9} T^{24} + 4914858 p^{9} T^{25} + 775926 p^{10} T^{26} + 89694 p^{11} T^{27} + 13275 p^{12} T^{28} + 1156 p^{13} T^{29} + 160 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 39 T + 32 p T^{2} + 16208 T^{3} + 230056 T^{4} + 2780632 T^{5} + 29604156 T^{6} + 283587370 T^{7} + 2485169966 T^{8} + 20154979173 T^{9} + 152664366097 T^{10} + 1087056903986 T^{11} + 7313935686038 T^{12} + 46666449986350 T^{13} + 283154391474588 T^{14} + 1636759340906011 T^{15} + 9023892718674587 T^{16} + 1636759340906011 p T^{17} + 283154391474588 p^{2} T^{18} + 46666449986350 p^{3} T^{19} + 7313935686038 p^{4} T^{20} + 1087056903986 p^{5} T^{21} + 152664366097 p^{6} T^{22} + 20154979173 p^{7} T^{23} + 2485169966 p^{8} T^{24} + 283587370 p^{9} T^{25} + 29604156 p^{10} T^{26} + 2780632 p^{11} T^{27} + 230056 p^{12} T^{28} + 16208 p^{13} T^{29} + 32 p^{15} T^{30} + 39 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 3 T + 255 T^{2} + 791 T^{3} + 32526 T^{4} + 101121 T^{5} + 2753424 T^{6} + 8468886 T^{7} + 174562090 T^{8} + 528154096 T^{9} + 8901907102 T^{10} + 26393054295 T^{11} + 382768477555 T^{12} + 1104116824605 T^{13} + 14294540542319 T^{14} + 1276700111171 p T^{15} + 470796472164880 T^{16} + 1276700111171 p^{2} T^{17} + 14294540542319 p^{2} T^{18} + 1104116824605 p^{3} T^{19} + 382768477555 p^{4} T^{20} + 26393054295 p^{5} T^{21} + 8901907102 p^{6} T^{22} + 528154096 p^{7} T^{23} + 174562090 p^{8} T^{24} + 8468886 p^{9} T^{25} + 2753424 p^{10} T^{26} + 101121 p^{11} T^{27} + 32526 p^{12} T^{28} + 791 p^{13} T^{29} + 255 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 28 T + 659 T^{2} + 11230 T^{3} + 168029 T^{4} + 2160736 T^{5} + 25241316 T^{6} + 267548264 T^{7} + 2629412336 T^{8} + 24030058442 T^{9} + 206141237141 T^{10} + 1665515462176 T^{11} + 12724942102618 T^{12} + 92203962435862 T^{13} + 634461000765648 T^{14} + 4155167012371534 T^{15} + 25893210569519688 T^{16} + 4155167012371534 p T^{17} + 634461000765648 p^{2} T^{18} + 92203962435862 p^{3} T^{19} + 12724942102618 p^{4} T^{20} + 1665515462176 p^{5} T^{21} + 206141237141 p^{6} T^{22} + 24030058442 p^{7} T^{23} + 2629412336 p^{8} T^{24} + 267548264 p^{9} T^{25} + 25241316 p^{10} T^{26} + 2160736 p^{11} T^{27} + 168029 p^{12} T^{28} + 11230 p^{13} T^{29} + 659 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 31 T + 845 T^{2} + 16144 T^{3} + 274419 T^{4} + 3937303 T^{5} + 51468151 T^{6} + 602401175 T^{7} + 6533515132 T^{8} + 65168277381 T^{9} + 609165944944 T^{10} + 5312237628797 T^{11} + 43729918253400 T^{12} + 338547209812076 T^{13} + 2484799560505024 T^{14} + 17223431233458923 T^{15} + 113409862522256899 T^{16} + 17223431233458923 p T^{17} + 2484799560505024 p^{2} T^{18} + 338547209812076 p^{3} T^{19} + 43729918253400 p^{4} T^{20} + 5312237628797 p^{5} T^{21} + 609165944944 p^{6} T^{22} + 65168277381 p^{7} T^{23} + 6533515132 p^{8} T^{24} + 602401175 p^{9} T^{25} + 51468151 p^{10} T^{26} + 3937303 p^{11} T^{27} + 274419 p^{12} T^{28} + 16144 p^{13} T^{29} + 845 p^{14} T^{30} + 31 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 5 T + 197 T^{2} - 481 T^{3} + 23722 T^{4} - 45125 T^{5} + 2315260 T^{6} - 3018587 T^{7} + 179938455 T^{8} - 163326585 T^{9} + 12170024380 T^{10} - 8524437672 T^{11} + 711460104377 T^{12} - 359409231654 T^{13} + 36587132230511 T^{14} - 16088275772689 T^{15} + 1675798794529458 T^{16} - 16088275772689 p T^{17} + 36587132230511 p^{2} T^{18} - 359409231654 p^{3} T^{19} + 711460104377 p^{4} T^{20} - 8524437672 p^{5} T^{21} + 12170024380 p^{6} T^{22} - 163326585 p^{7} T^{23} + 179938455 p^{8} T^{24} - 3018587 p^{9} T^{25} + 2315260 p^{10} T^{26} - 45125 p^{11} T^{27} + 23722 p^{12} T^{28} - 481 p^{13} T^{29} + 197 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + p T + 1521 T^{2} + 35695 T^{3} + 695344 T^{4} + 11443167 T^{5} + 166215515 T^{6} + 2150890354 T^{7} + 25336473439 T^{8} + 5813618469 p T^{9} + 2733546016518 T^{10} + 25462180643677 T^{11} + 222818061560779 T^{12} + 1835643848459763 T^{13} + 14325453981515050 T^{14} + 2254027551486834 p T^{15} + 745268954307368866 T^{16} + 2254027551486834 p^{2} T^{17} + 14325453981515050 p^{2} T^{18} + 1835643848459763 p^{3} T^{19} + 222818061560779 p^{4} T^{20} + 25462180643677 p^{5} T^{21} + 2733546016518 p^{6} T^{22} + 5813618469 p^{8} T^{23} + 25336473439 p^{8} T^{24} + 2150890354 p^{9} T^{25} + 166215515 p^{10} T^{26} + 11443167 p^{11} T^{27} + 695344 p^{12} T^{28} + 35695 p^{13} T^{29} + 1521 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 36 T + 1175 T^{2} + 26167 T^{3} + 526553 T^{4} + 8814464 T^{5} + 135423948 T^{6} + 1846719868 T^{7} + 23401307600 T^{8} + 271068847869 T^{9} + 2943032349289 T^{10} + 29666205870053 T^{11} + 281889792560510 T^{12} + 2508334485133314 T^{13} + 21112835718481295 T^{14} + 167153704971172239 T^{15} + 1253870241859978341 T^{16} + 167153704971172239 p T^{17} + 21112835718481295 p^{2} T^{18} + 2508334485133314 p^{3} T^{19} + 281889792560510 p^{4} T^{20} + 29666205870053 p^{5} T^{21} + 2943032349289 p^{6} T^{22} + 271068847869 p^{7} T^{23} + 23401307600 p^{8} T^{24} + 1846719868 p^{9} T^{25} + 135423948 p^{10} T^{26} + 8814464 p^{11} T^{27} + 526553 p^{12} T^{28} + 26167 p^{13} T^{29} + 1175 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 22 T + 726 T^{2} + 11788 T^{3} + 229971 T^{4} + 2989207 T^{5} + 44054827 T^{6} + 482781207 T^{7} + 5931009252 T^{8} + 57058208894 T^{9} + 619281836577 T^{10} + 5407165081688 T^{11} + 53761513199754 T^{12} + 435350040127477 T^{13} + 4029639992006474 T^{14} + 30479162475907733 T^{15} + 263314323349656692 T^{16} + 30479162475907733 p T^{17} + 4029639992006474 p^{2} T^{18} + 435350040127477 p^{3} T^{19} + 53761513199754 p^{4} T^{20} + 5407165081688 p^{5} T^{21} + 619281836577 p^{6} T^{22} + 57058208894 p^{7} T^{23} + 5931009252 p^{8} T^{24} + 482781207 p^{9} T^{25} + 44054827 p^{10} T^{26} + 2989207 p^{11} T^{27} + 229971 p^{12} T^{28} + 11788 p^{13} T^{29} + 726 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 4 T + 508 T^{2} - 486 T^{3} + 121581 T^{4} + 261148 T^{5} + 19273613 T^{6} + 99314231 T^{7} + 2421011347 T^{8} + 17808146326 T^{9} + 267088396674 T^{10} + 2154454341593 T^{11} + 26510155079768 T^{12} + 200163741901303 T^{13} + 2300638246781225 T^{14} + 15443017906683695 T^{15} + 168758840060292406 T^{16} + 15443017906683695 p T^{17} + 2300638246781225 p^{2} T^{18} + 200163741901303 p^{3} T^{19} + 26510155079768 p^{4} T^{20} + 2154454341593 p^{5} T^{21} + 267088396674 p^{6} T^{22} + 17808146326 p^{7} T^{23} + 2421011347 p^{8} T^{24} + 99314231 p^{9} T^{25} + 19273613 p^{10} T^{26} + 261148 p^{11} T^{27} + 121581 p^{12} T^{28} - 486 p^{13} T^{29} + 508 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 5 T + 707 T^{2} + 3199 T^{3} + 248382 T^{4} + 1037845 T^{5} + 57673238 T^{6} + 225467389 T^{7} + 9923775540 T^{8} + 36556228657 T^{9} + 1343843822580 T^{10} + 4670089592113 T^{11} + 148323449480723 T^{12} + 484195559141675 T^{13} + 13622265401829407 T^{14} + 41396926211404761 T^{15} + 1052601653972946460 T^{16} + 41396926211404761 p T^{17} + 13622265401829407 p^{2} T^{18} + 484195559141675 p^{3} T^{19} + 148323449480723 p^{4} T^{20} + 4670089592113 p^{5} T^{21} + 1343843822580 p^{6} T^{22} + 36556228657 p^{7} T^{23} + 9923775540 p^{8} T^{24} + 225467389 p^{9} T^{25} + 57673238 p^{10} T^{26} + 1037845 p^{11} T^{27} + 248382 p^{12} T^{28} + 3199 p^{13} T^{29} + 707 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 35 T + 1035 T^{2} + 20128 T^{3} + 344201 T^{4} + 4528714 T^{5} + 52898365 T^{6} + 472574263 T^{7} + 3575301535 T^{8} + 14472242265 T^{9} - 10259759258 T^{10} - 1413670638428 T^{11} - 13344230940186 T^{12} - 98999228688970 T^{13} + 14056056520030 T^{14} + 4087388912498153 T^{15} + 69628091750516746 T^{16} + 4087388912498153 p T^{17} + 14056056520030 p^{2} T^{18} - 98999228688970 p^{3} T^{19} - 13344230940186 p^{4} T^{20} - 1413670638428 p^{5} T^{21} - 10259759258 p^{6} T^{22} + 14472242265 p^{7} T^{23} + 3575301535 p^{8} T^{24} + 472574263 p^{9} T^{25} + 52898365 p^{10} T^{26} + 4528714 p^{11} T^{27} + 344201 p^{12} T^{28} + 20128 p^{13} T^{29} + 1035 p^{14} T^{30} + 35 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 48 T + 1716 T^{2} + 43733 T^{3} + 949683 T^{4} + 17347139 T^{5} + 284281513 T^{6} + 4151599168 T^{7} + 55961588988 T^{8} + 692772349249 T^{9} + 8070707342734 T^{10} + 88091360119644 T^{11} + 919032606186396 T^{12} + 9123573520975185 T^{13} + 87713721926917761 T^{14} + 811771645641552513 T^{15} + 7336995592941878765 T^{16} + 811771645641552513 p T^{17} + 87713721926917761 p^{2} T^{18} + 9123573520975185 p^{3} T^{19} + 919032606186396 p^{4} T^{20} + 88091360119644 p^{5} T^{21} + 8070707342734 p^{6} T^{22} + 692772349249 p^{7} T^{23} + 55961588988 p^{8} T^{24} + 4151599168 p^{9} T^{25} + 284281513 p^{10} T^{26} + 17347139 p^{11} T^{27} + 949683 p^{12} T^{28} + 43733 p^{13} T^{29} + 1716 p^{14} T^{30} + 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 42 T + 1656 T^{2} + 45425 T^{3} + 1121429 T^{4} + 23462170 T^{5} + 447315865 T^{6} + 7661927885 T^{7} + 121285518000 T^{8} + 1767394289294 T^{9} + 24044579507563 T^{10} + 304858972200110 T^{11} + 3632307914546706 T^{12} + 40607544102297339 T^{13} + 428241004835045976 T^{14} + 4252453091155187749 T^{15} + 39902891433324960408 T^{16} + 4252453091155187749 p T^{17} + 428241004835045976 p^{2} T^{18} + 40607544102297339 p^{3} T^{19} + 3632307914546706 p^{4} T^{20} + 304858972200110 p^{5} T^{21} + 24044579507563 p^{6} T^{22} + 1767394289294 p^{7} T^{23} + 121285518000 p^{8} T^{24} + 7661927885 p^{9} T^{25} + 447315865 p^{10} T^{26} + 23462170 p^{11} T^{27} + 1121429 p^{12} T^{28} + 45425 p^{13} T^{29} + 1656 p^{14} T^{30} + 42 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 20 T + 597 T^{2} - 8420 T^{3} + 158284 T^{4} - 1939051 T^{5} + 29478345 T^{6} - 339122373 T^{7} + 4443220742 T^{8} - 48716866782 T^{9} + 567385758156 T^{10} - 5969911399957 T^{11} + 63612180884481 T^{12} - 645883424308806 T^{13} + 6449935124831842 T^{14} - 63212766860278727 T^{15} + 599655028685899152 T^{16} - 63212766860278727 p T^{17} + 6449935124831842 p^{2} T^{18} - 645883424308806 p^{3} T^{19} + 63612180884481 p^{4} T^{20} - 5969911399957 p^{5} T^{21} + 567385758156 p^{6} T^{22} - 48716866782 p^{7} T^{23} + 4443220742 p^{8} T^{24} - 339122373 p^{9} T^{25} + 29478345 p^{10} T^{26} - 1939051 p^{11} T^{27} + 158284 p^{12} T^{28} - 8420 p^{13} T^{29} + 597 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 2 T + 621 T^{2} - 213 T^{3} + 205664 T^{4} - 437598 T^{5} + 49093119 T^{6} - 163737942 T^{7} + 9352075548 T^{8} - 38413772588 T^{9} + 1489316155695 T^{10} - 6708588826879 T^{11} + 203631678495233 T^{12} - 934785974558696 T^{13} + 24183348030921161 T^{14} - 107966335249417574 T^{15} + 2507492762870435228 T^{16} - 107966335249417574 p T^{17} + 24183348030921161 p^{2} T^{18} - 934785974558696 p^{3} T^{19} + 203631678495233 p^{4} T^{20} - 6708588826879 p^{5} T^{21} + 1489316155695 p^{6} T^{22} - 38413772588 p^{7} T^{23} + 9352075548 p^{8} T^{24} - 163737942 p^{9} T^{25} + 49093119 p^{10} T^{26} - 437598 p^{11} T^{27} + 205664 p^{12} T^{28} - 213 p^{13} T^{29} + 621 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.30353654542815962051557975014, −3.26324165716789478590745443506, −3.23626276081557395490977497786, −3.00091463121080336648513515386, −2.95357829517511742644497605647, −2.88962471129033408050024358715, −2.85794098813429330868440139789, −2.80116019531676023202934154915, −2.72353120612621954446739849017, −2.65591966457709086969978295114, −2.54339630373421989091933329504, −2.35274616343966691802482637932, −2.29529816020235326705233657726, −2.04327356620007429191694169137, −1.97483335730951043689896294454, −1.82510564369302480815894077480, −1.70400679602482255777587489213, −1.69646327185045402303115284470, −1.55243543766209378527612045820, −1.42643642647304201788396192526, −1.38967294234567652468835695019, −1.32080753287751811647151706440, −1.06951894810323582306156696495, −1.05375389875380089079224032697, −0.814946723971357906412223086233, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.814946723971357906412223086233, 1.05375389875380089079224032697, 1.06951894810323582306156696495, 1.32080753287751811647151706440, 1.38967294234567652468835695019, 1.42643642647304201788396192526, 1.55243543766209378527612045820, 1.69646327185045402303115284470, 1.70400679602482255777587489213, 1.82510564369302480815894077480, 1.97483335730951043689896294454, 2.04327356620007429191694169137, 2.29529816020235326705233657726, 2.35274616343966691802482637932, 2.54339630373421989091933329504, 2.65591966457709086969978295114, 2.72353120612621954446739849017, 2.80116019531676023202934154915, 2.85794098813429330868440139789, 2.88962471129033408050024358715, 2.95357829517511742644497605647, 3.00091463121080336648513515386, 3.23626276081557395490977497786, 3.26324165716789478590745443506, 3.30353654542815962051557975014
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.