Dirichlet series
| L(s) = 1 | − 16·3-s + 5·5-s − 4·7-s + 136·9-s + 5·11-s + 8·13-s − 80·15-s + 7·17-s + 19-s + 64·21-s + 16·23-s − 12·25-s − 816·27-s + 16·29-s − 2·31-s − 80·33-s − 20·35-s + 14·37-s − 128·39-s − 41-s − 13·43-s + 680·45-s − 4·47-s − 33·49-s − 112·51-s + 19·53-s + 25·55-s + ⋯ |
| L(s) = 1 | − 9.23·3-s + 2.23·5-s − 1.51·7-s + 45.3·9-s + 1.50·11-s + 2.21·13-s − 20.6·15-s + 1.69·17-s + 0.229·19-s + 13.9·21-s + 3.33·23-s − 2.39·25-s − 157.·27-s + 2.97·29-s − 0.359·31-s − 13.9·33-s − 3.38·35-s + 2.30·37-s − 20.4·39-s − 0.156·41-s − 1.98·43-s + 101.·45-s − 0.583·47-s − 4.71·49-s − 15.6·51-s + 2.60·53-s + 3.37·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.75080\times 10^{28}\) |
| Root analytic conductor: | \(7.99451\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 23^{16} \cdot 29^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9473774984\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9473774984\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{16} \) | |
| 23 | \( ( 1 - T )^{16} \) | |
| 29 | \( ( 1 - T )^{16} \) | |
| good | 5 | \( 1 - p T + 37 T^{2} - 141 T^{3} + 624 T^{4} - 1963 T^{5} + 6732 T^{6} - 18203 T^{7} + 52813 T^{8} - 998 p^{3} T^{9} + 319438 T^{10} - 668382 T^{11} + 1576116 T^{12} - 3007394 T^{13} + 6973497 T^{14} - 13024338 T^{15} + 32347212 T^{16} - 13024338 p T^{17} + 6973497 p^{2} T^{18} - 3007394 p^{3} T^{19} + 1576116 p^{4} T^{20} - 668382 p^{5} T^{21} + 319438 p^{6} T^{22} - 998 p^{10} T^{23} + 52813 p^{8} T^{24} - 18203 p^{9} T^{25} + 6732 p^{10} T^{26} - 1963 p^{11} T^{27} + 624 p^{12} T^{28} - 141 p^{13} T^{29} + 37 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 + 4 T + p^{2} T^{2} + 143 T^{3} + 1146 T^{4} + 2617 T^{5} + 17824 T^{6} + 31512 T^{7} + 209140 T^{8} + 272508 T^{9} + 1998130 T^{10} + 1731837 T^{11} + 2350874 p T^{12} + 8189843 T^{13} + 123459485 T^{14} + 33939848 T^{15} + 879401590 T^{16} + 33939848 p T^{17} + 123459485 p^{2} T^{18} + 8189843 p^{3} T^{19} + 2350874 p^{5} T^{20} + 1731837 p^{5} T^{21} + 1998130 p^{6} T^{22} + 272508 p^{7} T^{23} + 209140 p^{8} T^{24} + 31512 p^{9} T^{25} + 17824 p^{10} T^{26} + 2617 p^{11} T^{27} + 1146 p^{12} T^{28} + 143 p^{13} T^{29} + p^{16} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 5 T + 86 T^{2} - 332 T^{3} + 3444 T^{4} - 10852 T^{5} + 89725 T^{6} - 239870 T^{7} + 1769895 T^{8} - 4176547 T^{9} + 29063850 T^{10} - 63510903 T^{11} + 421824120 T^{12} - 883388327 T^{13} + 5513061075 T^{14} - 11073438454 T^{15} + 64368763384 T^{16} - 11073438454 p T^{17} + 5513061075 p^{2} T^{18} - 883388327 p^{3} T^{19} + 421824120 p^{4} T^{20} - 63510903 p^{5} T^{21} + 29063850 p^{6} T^{22} - 4176547 p^{7} T^{23} + 1769895 p^{8} T^{24} - 239870 p^{9} T^{25} + 89725 p^{10} T^{26} - 10852 p^{11} T^{27} + 3444 p^{12} T^{28} - 332 p^{13} T^{29} + 86 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 8 T + 88 T^{2} - 521 T^{3} + 3554 T^{4} - 17904 T^{5} + 95493 T^{6} - 419436 T^{7} + 1852540 T^{8} - 7049837 T^{9} + 26303780 T^{10} - 86286315 T^{11} + 274232175 T^{12} - 769412620 T^{13} + 2146397951 T^{14} - 5657924955 T^{15} + 18836760148 T^{16} - 5657924955 p T^{17} + 2146397951 p^{2} T^{18} - 769412620 p^{3} T^{19} + 274232175 p^{4} T^{20} - 86286315 p^{5} T^{21} + 26303780 p^{6} T^{22} - 7049837 p^{7} T^{23} + 1852540 p^{8} T^{24} - 419436 p^{9} T^{25} + 95493 p^{10} T^{26} - 17904 p^{11} T^{27} + 3554 p^{12} T^{28} - 521 p^{13} T^{29} + 88 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 7 T + 94 T^{2} - 516 T^{3} + 4635 T^{4} - 21224 T^{5} + 155819 T^{6} - 616477 T^{7} + 4125685 T^{8} - 14665513 T^{9} + 95186233 T^{10} - 319182182 T^{11} + 2045014477 T^{12} - 6632736614 T^{13} + 2407229918 p T^{14} - 127330577499 T^{15} + 739039136132 T^{16} - 127330577499 p T^{17} + 2407229918 p^{3} T^{18} - 6632736614 p^{3} T^{19} + 2045014477 p^{4} T^{20} - 319182182 p^{5} T^{21} + 95186233 p^{6} T^{22} - 14665513 p^{7} T^{23} + 4125685 p^{8} T^{24} - 616477 p^{9} T^{25} + 155819 p^{10} T^{26} - 21224 p^{11} T^{27} + 4635 p^{12} T^{28} - 516 p^{13} T^{29} + 94 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - T + 99 T^{2} - 272 T^{3} + 5907 T^{4} - 19503 T^{5} + 274410 T^{6} - 938905 T^{7} + 10142726 T^{8} - 35268703 T^{9} + 315815220 T^{10} - 1072284791 T^{11} + 444231895 p T^{12} - 27601647452 T^{13} + 195584026583 T^{14} - 607609573255 T^{15} + 3971249033986 T^{16} - 607609573255 p T^{17} + 195584026583 p^{2} T^{18} - 27601647452 p^{3} T^{19} + 444231895 p^{5} T^{20} - 1072284791 p^{5} T^{21} + 315815220 p^{6} T^{22} - 35268703 p^{7} T^{23} + 10142726 p^{8} T^{24} - 938905 p^{9} T^{25} + 274410 p^{10} T^{26} - 19503 p^{11} T^{27} + 5907 p^{12} T^{28} - 272 p^{13} T^{29} + 99 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 2 T + 304 T^{2} + 245 T^{3} + 44623 T^{4} - 11084 T^{5} + 4288800 T^{6} - 4847444 T^{7} + 307009964 T^{8} - 552289079 T^{9} + 17496878574 T^{10} - 39110100451 T^{11} + 822507319129 T^{12} - 2011025006762 T^{13} + 32509986488770 T^{14} - 2563723857303 p T^{15} + 1091583949781846 T^{16} - 2563723857303 p^{2} T^{17} + 32509986488770 p^{2} T^{18} - 2011025006762 p^{3} T^{19} + 822507319129 p^{4} T^{20} - 39110100451 p^{5} T^{21} + 17496878574 p^{6} T^{22} - 552289079 p^{7} T^{23} + 307009964 p^{8} T^{24} - 4847444 p^{9} T^{25} + 4288800 p^{10} T^{26} - 11084 p^{11} T^{27} + 44623 p^{12} T^{28} + 245 p^{13} T^{29} + 304 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 14 T + 433 T^{2} - 4927 T^{3} + 86193 T^{4} - 831169 T^{5} + 10708358 T^{6} - 89834264 T^{7} + 944503836 T^{8} - 7025048824 T^{9} + 63718267980 T^{10} - 427096455754 T^{11} + 3465679544791 T^{12} - 21266913245502 T^{13} + 158399291355657 T^{14} - 902356988343606 T^{15} + 6265265623498270 T^{16} - 902356988343606 p T^{17} + 158399291355657 p^{2} T^{18} - 21266913245502 p^{3} T^{19} + 3465679544791 p^{4} T^{20} - 427096455754 p^{5} T^{21} + 63718267980 p^{6} T^{22} - 7025048824 p^{7} T^{23} + 944503836 p^{8} T^{24} - 89834264 p^{9} T^{25} + 10708358 p^{10} T^{26} - 831169 p^{11} T^{27} + 86193 p^{12} T^{28} - 4927 p^{13} T^{29} + 433 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + T + 291 T^{2} - p T^{3} + 44852 T^{4} - 49667 T^{5} + 4812456 T^{6} - 9136611 T^{7} + 402888069 T^{8} - 986573010 T^{9} + 27907113690 T^{10} - 76364900726 T^{11} + 1647328324588 T^{12} - 4609023514894 T^{13} + 83871311547675 T^{14} - 226887864044812 T^{15} + 3695605203382292 T^{16} - 226887864044812 p T^{17} + 83871311547675 p^{2} T^{18} - 4609023514894 p^{3} T^{19} + 1647328324588 p^{4} T^{20} - 76364900726 p^{5} T^{21} + 27907113690 p^{6} T^{22} - 986573010 p^{7} T^{23} + 402888069 p^{8} T^{24} - 9136611 p^{9} T^{25} + 4812456 p^{10} T^{26} - 49667 p^{11} T^{27} + 44852 p^{12} T^{28} - p^{14} T^{29} + 291 p^{14} T^{30} + p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 13 T + 476 T^{2} + 5598 T^{3} + 110710 T^{4} + 1176341 T^{5} + 16654857 T^{6} + 160084609 T^{7} + 1816376364 T^{8} + 15837049807 T^{9} + 152933525073 T^{10} + 1214207439677 T^{11} + 10347972361250 T^{12} + 75132582499762 T^{13} + 577883827649346 T^{14} + 3850381644712395 T^{15} + 27062667776285686 T^{16} + 3850381644712395 p T^{17} + 577883827649346 p^{2} T^{18} + 75132582499762 p^{3} T^{19} + 10347972361250 p^{4} T^{20} + 1214207439677 p^{5} T^{21} + 152933525073 p^{6} T^{22} + 15837049807 p^{7} T^{23} + 1816376364 p^{8} T^{24} + 160084609 p^{9} T^{25} + 16654857 p^{10} T^{26} + 1176341 p^{11} T^{27} + 110710 p^{12} T^{28} + 5598 p^{13} T^{29} + 476 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 4 T + 350 T^{2} + 1492 T^{3} + 62253 T^{4} + 255333 T^{5} + 7303915 T^{6} + 26468619 T^{7} + 624450494 T^{8} + 1794433697 T^{9} + 41128444707 T^{10} + 77527415663 T^{11} + 2197732514771 T^{12} + 1679668521838 T^{13} + 103068758802868 T^{14} - 11943252911510 T^{15} + 4740210975040514 T^{16} - 11943252911510 p T^{17} + 103068758802868 p^{2} T^{18} + 1679668521838 p^{3} T^{19} + 2197732514771 p^{4} T^{20} + 77527415663 p^{5} T^{21} + 41128444707 p^{6} T^{22} + 1794433697 p^{7} T^{23} + 624450494 p^{8} T^{24} + 26468619 p^{9} T^{25} + 7303915 p^{10} T^{26} + 255333 p^{11} T^{27} + 62253 p^{12} T^{28} + 1492 p^{13} T^{29} + 350 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 19 T + 375 T^{2} - 3609 T^{3} + 777 p T^{4} - 228676 T^{5} + 2170379 T^{6} - 3229288 T^{7} + 70602930 T^{8} + 333959474 T^{9} + 4519420709 T^{10} + 13237540954 T^{11} + 461465245523 T^{12} + 36740101119 T^{13} + 31624213422425 T^{14} + 1919664742309 T^{15} + 1732957206033114 T^{16} + 1919664742309 p T^{17} + 31624213422425 p^{2} T^{18} + 36740101119 p^{3} T^{19} + 461465245523 p^{4} T^{20} + 13237540954 p^{5} T^{21} + 4519420709 p^{6} T^{22} + 333959474 p^{7} T^{23} + 70602930 p^{8} T^{24} - 3229288 p^{9} T^{25} + 2170379 p^{10} T^{26} - 228676 p^{11} T^{27} + 777 p^{13} T^{28} - 3609 p^{13} T^{29} + 375 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 12 T + 525 T^{2} - 5770 T^{3} + 134425 T^{4} - 1331890 T^{5} + 22452546 T^{6} - 200163974 T^{7} + 2774131676 T^{8} - 22471902618 T^{9} + 273551901484 T^{10} - 2043185035018 T^{11} + 22645172402319 T^{12} - 157768996900930 T^{13} + 27497583154439 p T^{14} - 10599336953451256 T^{15} + 101998757649670230 T^{16} - 10599336953451256 p T^{17} + 27497583154439 p^{3} T^{18} - 157768996900930 p^{3} T^{19} + 22645172402319 p^{4} T^{20} - 2043185035018 p^{5} T^{21} + 273551901484 p^{6} T^{22} - 22471902618 p^{7} T^{23} + 2774131676 p^{8} T^{24} - 200163974 p^{9} T^{25} + 22452546 p^{10} T^{26} - 1331890 p^{11} T^{27} + 134425 p^{12} T^{28} - 5770 p^{13} T^{29} + 525 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 21 T + 688 T^{2} - 10896 T^{3} + 204244 T^{4} - 2587358 T^{5} + 35323540 T^{6} - 370244264 T^{7} + 3999665515 T^{8} - 35396717507 T^{9} + 313490113918 T^{10} - 2378948179079 T^{11} + 17605136800576 T^{12} - 118427419531199 T^{13} + 770078149745886 T^{14} - 5245687475473816 T^{15} + 37051046350245696 T^{16} - 5245687475473816 p T^{17} + 770078149745886 p^{2} T^{18} - 118427419531199 p^{3} T^{19} + 17605136800576 p^{4} T^{20} - 2378948179079 p^{5} T^{21} + 313490113918 p^{6} T^{22} - 35396717507 p^{7} T^{23} + 3999665515 p^{8} T^{24} - 370244264 p^{9} T^{25} + 35323540 p^{10} T^{26} - 2587358 p^{11} T^{27} + 204244 p^{12} T^{28} - 10896 p^{13} T^{29} + 688 p^{14} T^{30} - 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 11 T + 677 T^{2} + 6253 T^{3} + 213651 T^{4} + 1742696 T^{5} + 42942399 T^{6} + 324003214 T^{7} + 6291846370 T^{8} + 45580026226 T^{9} + 724929048128 T^{10} + 5157895783324 T^{11} + 69004839961067 T^{12} + 484256922674237 T^{13} + 5621118295889744 T^{14} + 38295076389901251 T^{15} + 400888800091719174 T^{16} + 38295076389901251 p T^{17} + 5621118295889744 p^{2} T^{18} + 484256922674237 p^{3} T^{19} + 69004839961067 p^{4} T^{20} + 5157895783324 p^{5} T^{21} + 724929048128 p^{6} T^{22} + 45580026226 p^{7} T^{23} + 6291846370 p^{8} T^{24} + 324003214 p^{9} T^{25} + 42942399 p^{10} T^{26} + 1742696 p^{11} T^{27} + 213651 p^{12} T^{28} + 6253 p^{13} T^{29} + 677 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 7 T + 117 T^{2} - 291 T^{3} + 21452 T^{4} - 136910 T^{5} + 2415623 T^{6} - 7185203 T^{7} + 230286309 T^{8} - 1493262365 T^{9} + 24972357372 T^{10} - 94846172137 T^{11} + 1702596997046 T^{12} - 10893914862268 T^{13} + 169597232593544 T^{14} - 750168218524879 T^{15} + 9606859347730640 T^{16} - 750168218524879 p T^{17} + 169597232593544 p^{2} T^{18} - 10893914862268 p^{3} T^{19} + 1702596997046 p^{4} T^{20} - 94846172137 p^{5} T^{21} + 24972357372 p^{6} T^{22} - 1493262365 p^{7} T^{23} + 230286309 p^{8} T^{24} - 7185203 p^{9} T^{25} + 2415623 p^{10} T^{26} - 136910 p^{11} T^{27} + 21452 p^{12} T^{28} - 291 p^{13} T^{29} + 117 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 13 T + 591 T^{2} + 6898 T^{3} + 182372 T^{4} + 1956529 T^{5} + 38630719 T^{6} + 383102958 T^{7} + 6243144468 T^{8} + 57513478640 T^{9} + 813550203641 T^{10} + 6978310738401 T^{11} + 88227180423132 T^{12} + 705198286076836 T^{13} + 8114168841062969 T^{14} + 60341860895801245 T^{15} + 639064450484286678 T^{16} + 60341860895801245 p T^{17} + 8114168841062969 p^{2} T^{18} + 705198286076836 p^{3} T^{19} + 88227180423132 p^{4} T^{20} + 6978310738401 p^{5} T^{21} + 813550203641 p^{6} T^{22} + 57513478640 p^{7} T^{23} + 6243144468 p^{8} T^{24} + 383102958 p^{9} T^{25} + 38630719 p^{10} T^{26} + 1956529 p^{11} T^{27} + 182372 p^{12} T^{28} + 6898 p^{13} T^{29} + 591 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 18 T + 579 T^{2} + 7553 T^{3} + 148040 T^{4} + 1700431 T^{5} + 26248023 T^{6} + 288677003 T^{7} + 3732561742 T^{8} + 39699718571 T^{9} + 446773746969 T^{10} + 4618561571973 T^{11} + 47031233415920 T^{12} + 468957869663831 T^{13} + 4404948383830349 T^{14} + 41771965462510402 T^{15} + 367448687940717810 T^{16} + 41771965462510402 p T^{17} + 4404948383830349 p^{2} T^{18} + 468957869663831 p^{3} T^{19} + 47031233415920 p^{4} T^{20} + 4618561571973 p^{5} T^{21} + 446773746969 p^{6} T^{22} + 39699718571 p^{7} T^{23} + 3732561742 p^{8} T^{24} + 288677003 p^{9} T^{25} + 26248023 p^{10} T^{26} + 1700431 p^{11} T^{27} + 148040 p^{12} T^{28} + 7553 p^{13} T^{29} + 579 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 25 T + 819 T^{2} - 14447 T^{3} + 292981 T^{4} - 4179204 T^{5} + 66556473 T^{6} - 823298784 T^{7} + 11268998106 T^{8} - 125689558730 T^{9} + 1544698676427 T^{10} - 15870363430370 T^{11} + 179127412519771 T^{12} - 1714485105914409 T^{13} + 18000417951575385 T^{14} - 161522367046609711 T^{15} + 1589621530804707978 T^{16} - 161522367046609711 p T^{17} + 18000417951575385 p^{2} T^{18} - 1714485105914409 p^{3} T^{19} + 179127412519771 p^{4} T^{20} - 15870363430370 p^{5} T^{21} + 1544698676427 p^{6} T^{22} - 125689558730 p^{7} T^{23} + 11268998106 p^{8} T^{24} - 823298784 p^{9} T^{25} + 66556473 p^{10} T^{26} - 4179204 p^{11} T^{27} + 292981 p^{12} T^{28} - 14447 p^{13} T^{29} + 819 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 12 T + 646 T^{2} - 4091 T^{3} + 177301 T^{4} - 406466 T^{5} + 32347743 T^{6} + 27803907 T^{7} + 4909962075 T^{8} + 13752862353 T^{9} + 670323428561 T^{10} + 2421001814046 T^{11} + 80440456506451 T^{12} + 327695186971247 T^{13} + 8347803991668250 T^{14} + 36640041474876984 T^{15} + 775716976754939256 T^{16} + 36640041474876984 p T^{17} + 8347803991668250 p^{2} T^{18} + 327695186971247 p^{3} T^{19} + 80440456506451 p^{4} T^{20} + 2421001814046 p^{5} T^{21} + 670323428561 p^{6} T^{22} + 13752862353 p^{7} T^{23} + 4909962075 p^{8} T^{24} + 27803907 p^{9} T^{25} + 32347743 p^{10} T^{26} - 406466 p^{11} T^{27} + 177301 p^{12} T^{28} - 4091 p^{13} T^{29} + 646 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 5 T + 746 T^{2} - 578 T^{3} + 251833 T^{4} + 959647 T^{5} + 54904494 T^{6} + 469001690 T^{7} + 9616658962 T^{8} + 114862667852 T^{9} + 1527018625282 T^{10} + 19504016809643 T^{11} + 218535622457863 T^{12} + 2629684761523704 T^{13} + 26972873110330406 T^{14} + 298659612495176735 T^{15} + 2833262019648399578 T^{16} + 298659612495176735 p T^{17} + 26972873110330406 p^{2} T^{18} + 2629684761523704 p^{3} T^{19} + 218535622457863 p^{4} T^{20} + 19504016809643 p^{5} T^{21} + 1527018625282 p^{6} T^{22} + 114862667852 p^{7} T^{23} + 9616658962 p^{8} T^{24} + 469001690 p^{9} T^{25} + 54904494 p^{10} T^{26} + 959647 p^{11} T^{27} + 251833 p^{12} T^{28} - 578 p^{13} T^{29} + 746 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−1.66877932542099159593341759602, −1.64969236014087390588551429415, −1.64214973986750483602864610259, −1.62003339473561120201652171924, −1.61688535608208590588788063930, −1.50363222793748032125586744593, −1.47389304756662621446749955226, −1.44035737178079189242228739008, −1.26614206233050415912771756557, −1.21068728938508431452046113396, −1.01914591021815005318196023993, −0.937894510332341865043757388756, −0.855558573664125524571347207579, −0.76975637896183821871004633539, −0.73560704391702737636059225823, −0.69038373908941894931825389365, −0.64384141394601771768198564137, −0.63752559154714647095328542268, −0.60568839191446078091643253597, −0.54370007951976906065920448198, −0.50699869374103678000533268577, −0.49903298795551450653559208768, −0.37501078171997851013496673400, −0.20151237865175211205558153971, −0.03987512946949730057391183493, 0.03987512946949730057391183493, 0.20151237865175211205558153971, 0.37501078171997851013496673400, 0.49903298795551450653559208768, 0.50699869374103678000533268577, 0.54370007951976906065920448198, 0.60568839191446078091643253597, 0.63752559154714647095328542268, 0.64384141394601771768198564137, 0.69038373908941894931825389365, 0.73560704391702737636059225823, 0.76975637896183821871004633539, 0.855558573664125524571347207579, 0.937894510332341865043757388756, 1.01914591021815005318196023993, 1.21068728938508431452046113396, 1.26614206233050415912771756557, 1.44035737178079189242228739008, 1.47389304756662621446749955226, 1.50363222793748032125586744593, 1.61688535608208590588788063930, 1.62003339473561120201652171924, 1.64214973986750483602864610259, 1.64969236014087390588551429415, 1.66877932542099159593341759602
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.