Dirichlet series
| L(s) = 1 | − 4·2-s − 16·3-s − 2·4-s + 12·5-s + 64·6-s + 28·8-s + 136·9-s − 48·10-s − 4·11-s + 32·12-s − 192·15-s − 20·16-s + 8·17-s − 544·18-s − 4·19-s − 24·20-s + 16·22-s − 12·23-s − 448·24-s + 28·25-s − 816·27-s − 16·29-s + 768·30-s + 4·31-s − 80·32-s + 64·33-s − 32·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 9.23·3-s − 4-s + 5.36·5-s + 26.1·6-s + 9.89·8-s + 45.3·9-s − 15.1·10-s − 1.20·11-s + 9.23·12-s − 49.5·15-s − 5·16-s + 1.94·17-s − 128.·18-s − 0.917·19-s − 5.36·20-s + 3.41·22-s − 2.50·23-s − 91.4·24-s + 28/5·25-s − 157.·27-s − 2.97·29-s + 140.·30-s + 0.718·31-s − 14.1·32-s + 11.1·33-s − 5.48·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 7^{32} \cdot 41^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.28042\times 10^{26}\) |
| Root analytic conductor: | \(6.93727\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 3^{16} \cdot 7^{32} \cdot 41^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
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 | 3 | \( ( 1 + T )^{16} \) |
| 7 | \( 1 \) | |
| 41 | \( ( 1 + T )^{16} \) | |
| good | 2 | \( 1 + p^{2} T + 9 p T^{2} + 13 p^{2} T^{3} + 19 p^{3} T^{4} + 23 p^{4} T^{5} + 433 p T^{6} + 461 p^{2} T^{7} + 471 p^{3} T^{8} + 1807 p^{2} T^{9} + 6645 p T^{10} + 727 p^{5} T^{11} + 9799 p^{2} T^{12} + 1973 p^{5} T^{13} + 24595 p^{2} T^{14} + 9163 p^{4} T^{15} + 212031 T^{16} + 9163 p^{5} T^{17} + 24595 p^{4} T^{18} + 1973 p^{8} T^{19} + 9799 p^{6} T^{20} + 727 p^{10} T^{21} + 6645 p^{7} T^{22} + 1807 p^{9} T^{23} + 471 p^{11} T^{24} + 461 p^{11} T^{25} + 433 p^{11} T^{26} + 23 p^{15} T^{27} + 19 p^{15} T^{28} + 13 p^{15} T^{29} + 9 p^{15} T^{30} + p^{17} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 - 12 T + 116 T^{2} - 32 p^{2} T^{3} + 958 p T^{4} - 24268 T^{5} + 4436 p^{2} T^{6} - 453432 T^{7} + 1709536 T^{8} - 5921964 T^{9} + 19182808 T^{10} - 57964364 T^{11} + 165304396 T^{12} - 443829604 T^{13} + 1131277364 T^{14} - 545767388 p T^{15} + 6267117097 T^{16} - 545767388 p^{2} T^{17} + 1131277364 p^{2} T^{18} - 443829604 p^{3} T^{19} + 165304396 p^{4} T^{20} - 57964364 p^{5} T^{21} + 19182808 p^{6} T^{22} - 5921964 p^{7} T^{23} + 1709536 p^{8} T^{24} - 453432 p^{9} T^{25} + 4436 p^{12} T^{26} - 24268 p^{11} T^{27} + 958 p^{13} T^{28} - 32 p^{15} T^{29} + 116 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 4 T + 96 T^{2} + 400 T^{3} + 4516 T^{4} + 18768 T^{5} + 137740 T^{6} + 553324 T^{7} + 3044188 T^{8} + 11588124 T^{9} + 52049524 T^{10} + 186082592 T^{11} + 728080508 T^{12} + 2455353280 T^{13} + 8863000560 T^{14} + 28708079668 T^{15} + 99884859990 T^{16} + 28708079668 p T^{17} + 8863000560 p^{2} T^{18} + 2455353280 p^{3} T^{19} + 728080508 p^{4} T^{20} + 186082592 p^{5} T^{21} + 52049524 p^{6} T^{22} + 11588124 p^{7} T^{23} + 3044188 p^{8} T^{24} + 553324 p^{9} T^{25} + 137740 p^{10} T^{26} + 18768 p^{11} T^{27} + 4516 p^{12} T^{28} + 400 p^{13} T^{29} + 96 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 124 T^{2} - 88 T^{3} + 7580 T^{4} - 10260 T^{5} + 306166 T^{6} - 587676 T^{7} + 9198200 T^{8} - 21918452 T^{9} + 218887020 T^{10} - 593410276 T^{11} + 329562534 p T^{12} - 946306032 p T^{13} + 70590008544 T^{14} - 200749663100 T^{15} + 991722771327 T^{16} - 200749663100 p T^{17} + 70590008544 p^{2} T^{18} - 946306032 p^{4} T^{19} + 329562534 p^{5} T^{20} - 593410276 p^{5} T^{21} + 218887020 p^{6} T^{22} - 21918452 p^{7} T^{23} + 9198200 p^{8} T^{24} - 587676 p^{9} T^{25} + 306166 p^{10} T^{26} - 10260 p^{11} T^{27} + 7580 p^{12} T^{28} - 88 p^{13} T^{29} + 124 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 8 T + 186 T^{2} - 1336 T^{3} + 17166 T^{4} - 109272 T^{5} + 1025802 T^{6} - 5815764 T^{7} + 44314300 T^{8} - 225732532 T^{9} + 1473725454 T^{10} - 6804736544 T^{11} + 39319757210 T^{12} - 165854015344 T^{13} + 50932820146 p T^{14} - 3353932447992 T^{15} + 16013803128506 T^{16} - 3353932447992 p T^{17} + 50932820146 p^{3} T^{18} - 165854015344 p^{3} T^{19} + 39319757210 p^{4} T^{20} - 6804736544 p^{5} T^{21} + 1473725454 p^{6} T^{22} - 225732532 p^{7} T^{23} + 44314300 p^{8} T^{24} - 5815764 p^{9} T^{25} + 1025802 p^{10} T^{26} - 109272 p^{11} T^{27} + 17166 p^{12} T^{28} - 1336 p^{13} T^{29} + 186 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 4 T + 142 T^{2} + 28 p T^{3} + 10634 T^{4} + 37952 T^{5} + 553528 T^{6} + 1904680 T^{7} + 22310056 T^{8} + 74178752 T^{9} + 734100868 T^{10} + 2348933536 T^{11} + 20308487778 T^{12} + 62022285764 T^{13} + 480541509318 T^{14} + 1384891924268 T^{15} + 9813842272122 T^{16} + 1384891924268 p T^{17} + 480541509318 p^{2} T^{18} + 62022285764 p^{3} T^{19} + 20308487778 p^{4} T^{20} + 2348933536 p^{5} T^{21} + 734100868 p^{6} T^{22} + 74178752 p^{7} T^{23} + 22310056 p^{8} T^{24} + 1904680 p^{9} T^{25} + 553528 p^{10} T^{26} + 37952 p^{11} T^{27} + 10634 p^{12} T^{28} + 28 p^{14} T^{29} + 142 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 12 T + 300 T^{2} + 2888 T^{3} + 41488 T^{4} + 336860 T^{5} + 3594064 T^{6} + 25379288 T^{7} + 221508684 T^{8} + 1388094300 T^{9} + 10408028028 T^{10} + 58672657936 T^{11} + 388802450036 T^{12} + 1988966597832 T^{13} + 11849558130646 T^{14} + 55257243205592 T^{15} + 298889278747841 T^{16} + 55257243205592 p T^{17} + 11849558130646 p^{2} T^{18} + 1988966597832 p^{3} T^{19} + 388802450036 p^{4} T^{20} + 58672657936 p^{5} T^{21} + 10408028028 p^{6} T^{22} + 1388094300 p^{7} T^{23} + 221508684 p^{8} T^{24} + 25379288 p^{9} T^{25} + 3594064 p^{10} T^{26} + 336860 p^{11} T^{27} + 41488 p^{12} T^{28} + 2888 p^{13} T^{29} + 300 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 16 T + 14 p T^{2} + 4976 T^{3} + 72870 T^{4} + 725544 T^{5} + 7877834 T^{6} + 66129352 T^{7} + 585112630 T^{8} + 4254752904 T^{9} + 32285362608 T^{10} + 208155513460 T^{11} + 48390664782 p T^{12} + 8203365860624 T^{13} + 50525131441338 T^{14} + 273195919660364 T^{15} + 1566678823311677 T^{16} + 273195919660364 p T^{17} + 50525131441338 p^{2} T^{18} + 8203365860624 p^{3} T^{19} + 48390664782 p^{5} T^{20} + 208155513460 p^{5} T^{21} + 32285362608 p^{6} T^{22} + 4254752904 p^{7} T^{23} + 585112630 p^{8} T^{24} + 66129352 p^{9} T^{25} + 7877834 p^{10} T^{26} + 725544 p^{11} T^{27} + 72870 p^{12} T^{28} + 4976 p^{13} T^{29} + 14 p^{15} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 4 T + 232 T^{2} - 272 T^{3} + 23624 T^{4} + 32792 T^{5} + 52414 p T^{6} + 5378348 T^{7} + 95038424 T^{8} + 392446940 T^{9} + 4855014770 T^{10} + 21125732288 T^{11} + 211674513816 T^{12} + 924087272744 T^{13} + 8061046899332 T^{14} + 33382580342364 T^{15} + 268840782252178 T^{16} + 33382580342364 p T^{17} + 8061046899332 p^{2} T^{18} + 924087272744 p^{3} T^{19} + 211674513816 p^{4} T^{20} + 21125732288 p^{5} T^{21} + 4855014770 p^{6} T^{22} + 392446940 p^{7} T^{23} + 95038424 p^{8} T^{24} + 5378348 p^{9} T^{25} + 52414 p^{11} T^{26} + 32792 p^{11} T^{27} + 23624 p^{12} T^{28} - 272 p^{13} T^{29} + 232 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 48 T + 1416 T^{2} + 30728 T^{3} + 541936 T^{4} + 8109248 T^{5} + 106223504 T^{6} + 1241193156 T^{7} + 13134389922 T^{8} + 127223966996 T^{9} + 1138424823192 T^{10} + 9477672586424 T^{11} + 73876900095216 T^{12} + 541868352358408 T^{13} + 3755924503975212 T^{14} + 24675921878913492 T^{15} + 153973622531243819 T^{16} + 24675921878913492 p T^{17} + 3755924503975212 p^{2} T^{18} + 541868352358408 p^{3} T^{19} + 73876900095216 p^{4} T^{20} + 9477672586424 p^{5} T^{21} + 1138424823192 p^{6} T^{22} + 127223966996 p^{7} T^{23} + 13134389922 p^{8} T^{24} + 1241193156 p^{9} T^{25} + 106223504 p^{10} T^{26} + 8109248 p^{11} T^{27} + 541936 p^{12} T^{28} + 30728 p^{13} T^{29} + 1416 p^{14} T^{30} + 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 16 T + 472 T^{2} + 6080 T^{3} + 101832 T^{4} + 1106076 T^{5} + 13658036 T^{6} + 129557424 T^{7} + 1305936920 T^{8} + 11133477344 T^{9} + 2253297160 p T^{10} + 759686130716 T^{11} + 5919283650060 T^{12} + 43351759871712 T^{13} + 309810122798232 T^{14} + 2133248545445288 T^{15} + 14193311115214706 T^{16} + 2133248545445288 p T^{17} + 309810122798232 p^{2} T^{18} + 43351759871712 p^{3} T^{19} + 5919283650060 p^{4} T^{20} + 759686130716 p^{5} T^{21} + 2253297160 p^{7} T^{22} + 11133477344 p^{7} T^{23} + 1305936920 p^{8} T^{24} + 129557424 p^{9} T^{25} + 13658036 p^{10} T^{26} + 1106076 p^{11} T^{27} + 101832 p^{12} T^{28} + 6080 p^{13} T^{29} + 472 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 36 T + 970 T^{2} - 19112 T^{3} + 323988 T^{4} - 4732780 T^{5} + 62562642 T^{6} - 750901968 T^{7} + 8350294552 T^{8} - 86291191188 T^{9} + 837015131308 T^{10} - 7638200263788 T^{11} + 65931028054248 T^{12} - 539058712093292 T^{13} + 4187042836363508 T^{14} - 30913110714099580 T^{15} + 217255841691050453 T^{16} - 30913110714099580 p T^{17} + 4187042836363508 p^{2} T^{18} - 539058712093292 p^{3} T^{19} + 65931028054248 p^{4} T^{20} - 7638200263788 p^{5} T^{21} + 837015131308 p^{6} T^{22} - 86291191188 p^{7} T^{23} + 8350294552 p^{8} T^{24} - 750901968 p^{9} T^{25} + 62562642 p^{10} T^{26} - 4732780 p^{11} T^{27} + 323988 p^{12} T^{28} - 19112 p^{13} T^{29} + 970 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 60 T + 38 p T^{2} + 48452 T^{3} + 926840 T^{4} + 14904060 T^{5} + 208733312 T^{6} + 2610003712 T^{7} + 29692405986 T^{8} + 312000586468 T^{9} + 3065607281316 T^{10} + 28442796034140 T^{11} + 250997249630834 T^{12} + 2116696293471160 T^{13} + 17102229807589320 T^{14} + 132522275383714976 T^{15} + 985054629185327913 T^{16} + 132522275383714976 p T^{17} + 17102229807589320 p^{2} T^{18} + 2116696293471160 p^{3} T^{19} + 250997249630834 p^{4} T^{20} + 28442796034140 p^{5} T^{21} + 3065607281316 p^{6} T^{22} + 312000586468 p^{7} T^{23} + 29692405986 p^{8} T^{24} + 2610003712 p^{9} T^{25} + 208733312 p^{10} T^{26} + 14904060 p^{11} T^{27} + 926840 p^{12} T^{28} + 48452 p^{13} T^{29} + 38 p^{15} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 36 T + 1238 T^{2} - 28056 T^{3} + 590152 T^{4} - 10124732 T^{5} + 162272994 T^{6} - 2279316140 T^{7} + 30153633728 T^{8} - 361420991004 T^{9} + 4106010948930 T^{10} - 43017709713236 T^{11} + 429063702203260 T^{12} - 3985426065777760 T^{13} + 35339903057493066 T^{14} - 293303526061749276 T^{15} + 2326887311250212714 T^{16} - 293303526061749276 p T^{17} + 35339903057493066 p^{2} T^{18} - 3985426065777760 p^{3} T^{19} + 429063702203260 p^{4} T^{20} - 43017709713236 p^{5} T^{21} + 4106010948930 p^{6} T^{22} - 361420991004 p^{7} T^{23} + 30153633728 p^{8} T^{24} - 2279316140 p^{9} T^{25} + 162272994 p^{10} T^{26} - 10124732 p^{11} T^{27} + 590152 p^{12} T^{28} - 28056 p^{13} T^{29} + 1238 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 4 T + 518 T^{2} - 2448 T^{3} + 135132 T^{4} - 712768 T^{5} + 23689024 T^{6} - 133999716 T^{7} + 3137221444 T^{8} - 18450841316 T^{9} + 333995956716 T^{10} - 1989279192272 T^{11} + 29660372180160 T^{12} - 174418156061488 T^{13} + 2247519933539238 T^{14} - 12697313124053556 T^{15} + 147175014200284562 T^{16} - 12697313124053556 p T^{17} + 2247519933539238 p^{2} T^{18} - 174418156061488 p^{3} T^{19} + 29660372180160 p^{4} T^{20} - 1989279192272 p^{5} T^{21} + 333995956716 p^{6} T^{22} - 18450841316 p^{7} T^{23} + 3137221444 p^{8} T^{24} - 133999716 p^{9} T^{25} + 23689024 p^{10} T^{26} - 712768 p^{11} T^{27} + 135132 p^{12} T^{28} - 2448 p^{13} T^{29} + 518 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 52 T + 1586 T^{2} + 35740 T^{3} + 668670 T^{4} + 10926900 T^{5} + 160967438 T^{6} + 2173943892 T^{7} + 27279832966 T^{8} + 320771579536 T^{9} + 3558878336944 T^{10} + 37421476641872 T^{11} + 374379289231312 T^{12} + 3572577984865256 T^{13} + 32589548005334360 T^{14} + 284513763061068648 T^{15} + 2379344682498376231 T^{16} + 284513763061068648 p T^{17} + 32589548005334360 p^{2} T^{18} + 3572577984865256 p^{3} T^{19} + 374379289231312 p^{4} T^{20} + 37421476641872 p^{5} T^{21} + 3558878336944 p^{6} T^{22} + 320771579536 p^{7} T^{23} + 27279832966 p^{8} T^{24} + 2173943892 p^{9} T^{25} + 160967438 p^{10} T^{26} + 10926900 p^{11} T^{27} + 668670 p^{12} T^{28} + 35740 p^{13} T^{29} + 1586 p^{14} T^{30} + 52 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 12 T + 524 T^{2} + 6144 T^{3} + 142430 T^{4} + 1534308 T^{5} + 25854030 T^{6} + 250867984 T^{7} + 3472823680 T^{8} + 30340412056 T^{9} + 368638078398 T^{10} + 2930308044412 T^{11} + 32733069611826 T^{12} + 241242807800808 T^{13} + 2572352535405856 T^{14} + 18031197716641476 T^{15} + 187679025498718762 T^{16} + 18031197716641476 p T^{17} + 2572352535405856 p^{2} T^{18} + 241242807800808 p^{3} T^{19} + 32733069611826 p^{4} T^{20} + 2930308044412 p^{5} T^{21} + 368638078398 p^{6} T^{22} + 30340412056 p^{7} T^{23} + 3472823680 p^{8} T^{24} + 250867984 p^{9} T^{25} + 25854030 p^{10} T^{26} + 1534308 p^{11} T^{27} + 142430 p^{12} T^{28} + 6144 p^{13} T^{29} + 524 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 16 T + 530 T^{2} - 5692 T^{3} + 117508 T^{4} - 915052 T^{5} + 16492920 T^{6} - 99072648 T^{7} + 1882460068 T^{8} - 9486627168 T^{9} + 196417527108 T^{10} - 866565200788 T^{11} + 18368497531236 T^{12} - 69153174751508 T^{13} + 1502837436663074 T^{14} - 4886985184019752 T^{15} + 112456480948156330 T^{16} - 4886985184019752 p T^{17} + 1502837436663074 p^{2} T^{18} - 69153174751508 p^{3} T^{19} + 18368497531236 p^{4} T^{20} - 866565200788 p^{5} T^{21} + 196417527108 p^{6} T^{22} - 9486627168 p^{7} T^{23} + 1882460068 p^{8} T^{24} - 99072648 p^{9} T^{25} + 16492920 p^{10} T^{26} - 915052 p^{11} T^{27} + 117508 p^{12} T^{28} - 5692 p^{13} T^{29} + 530 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 36 T + 1300 T^{2} + 29640 T^{3} + 644442 T^{4} + 11062384 T^{5} + 181855802 T^{6} + 2555579240 T^{7} + 34854549406 T^{8} + 423583775828 T^{9} + 5067792244106 T^{10} + 55382235796288 T^{11} + 601566136335572 T^{12} + 6041847061296648 T^{13} + 60445891226952254 T^{14} + 560897195446031972 T^{15} + 5176511227581737867 T^{16} + 560897195446031972 p T^{17} + 60445891226952254 p^{2} T^{18} + 6041847061296648 p^{3} T^{19} + 601566136335572 p^{4} T^{20} + 55382235796288 p^{5} T^{21} + 5067792244106 p^{6} T^{22} + 423583775828 p^{7} T^{23} + 34854549406 p^{8} T^{24} + 2555579240 p^{9} T^{25} + 181855802 p^{10} T^{26} + 11062384 p^{11} T^{27} + 644442 p^{12} T^{28} + 29640 p^{13} T^{29} + 1300 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 32 T + 1396 T^{2} - 31864 T^{3} + 825548 T^{4} - 15014192 T^{5} + 293221524 T^{6} - 4489655592 T^{7} + 72121934324 T^{8} - 959107564728 T^{9} + 13229625241788 T^{10} - 155674912147360 T^{11} + 1888666986914388 T^{12} - 19883593515344968 T^{13} + 215145806882517948 T^{14} - 2038137667495716048 T^{15} + 19817106231075561046 T^{16} - 2038137667495716048 p T^{17} + 215145806882517948 p^{2} T^{18} - 19883593515344968 p^{3} T^{19} + 1888666986914388 p^{4} T^{20} - 155674912147360 p^{5} T^{21} + 13229625241788 p^{6} T^{22} - 959107564728 p^{7} T^{23} + 72121934324 p^{8} T^{24} - 4489655592 p^{9} T^{25} + 293221524 p^{10} T^{26} - 15014192 p^{11} T^{27} + 825548 p^{12} T^{28} - 31864 p^{13} T^{29} + 1396 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 12 T + 666 T^{2} - 7704 T^{3} + 229332 T^{4} - 2363360 T^{5} + 51092866 T^{6} - 453433084 T^{7} + 8008291596 T^{8} - 58337333364 T^{9} + 916755309062 T^{10} - 4982685787248 T^{11} + 78585958110684 T^{12} - 250127242060344 T^{13} + 5471322997582174 T^{14} - 4863344876453860 T^{15} + 410068025144719350 T^{16} - 4863344876453860 p T^{17} + 5471322997582174 p^{2} T^{18} - 250127242060344 p^{3} T^{19} + 78585958110684 p^{4} T^{20} - 4982685787248 p^{5} T^{21} + 916755309062 p^{6} T^{22} - 58337333364 p^{7} T^{23} + 8008291596 p^{8} T^{24} - 453433084 p^{9} T^{25} + 51092866 p^{10} T^{26} - 2363360 p^{11} T^{27} + 229332 p^{12} T^{28} - 7704 p^{13} T^{29} + 666 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 48 T + 1860 T^{2} + 50420 T^{3} + 1195958 T^{4} + 23640396 T^{5} + 423939732 T^{6} + 6708542068 T^{7} + 98264686644 T^{8} + 1305241581368 T^{9} + 16292429709952 T^{10} + 187916792241016 T^{11} + 2072730202551542 T^{12} + 21569831957945712 T^{13} + 220137841603022498 T^{14} + 2175169388860070712 T^{15} + 21616316654655447003 T^{16} + 2175169388860070712 p T^{17} + 220137841603022498 p^{2} T^{18} + 21569831957945712 p^{3} T^{19} + 2072730202551542 p^{4} T^{20} + 187916792241016 p^{5} T^{21} + 16292429709952 p^{6} T^{22} + 1305241581368 p^{7} T^{23} + 98264686644 p^{8} T^{24} + 6708542068 p^{9} T^{25} + 423939732 p^{10} T^{26} + 23640396 p^{11} T^{27} + 1195958 p^{12} T^{28} + 50420 p^{13} T^{29} + 1860 p^{14} T^{30} + 48 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.16221263729052953047964067842, −2.11519299915871821486560519638, −2.06163842739989949553834581914, −1.99341598981168588209902825551, −1.98680973677376612674451772104, −1.96399745940612443252913534188, −1.92293739266139938144572286158, −1.89907671372961254683505791443, −1.73627937499180000444989750799, −1.64623460972232894223178352861, −1.60251157065738642866461982067, −1.58437385479487271688920847776, −1.51269757607961790263588855576, −1.40865716418426768386447537870, −1.25536569590324131901159666262, −1.25451133973638491739122497524, −1.24157544235774194989864210816, −1.22636541358984636099511997840, −1.15774374973047577508202631392, −1.12859207534709388588392320592, −1.04464078028975173881987817690, −1.02909455695036978810109897699, −1.02359699599463388018159897127, −0.956283294999502183027472815913, −0.77997056771773161078763489370, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77997056771773161078763489370, 0.956283294999502183027472815913, 1.02359699599463388018159897127, 1.02909455695036978810109897699, 1.04464078028975173881987817690, 1.12859207534709388588392320592, 1.15774374973047577508202631392, 1.22636541358984636099511997840, 1.24157544235774194989864210816, 1.25451133973638491739122497524, 1.25536569590324131901159666262, 1.40865716418426768386447537870, 1.51269757607961790263588855576, 1.58437385479487271688920847776, 1.60251157065738642866461982067, 1.64623460972232894223178352861, 1.73627937499180000444989750799, 1.89907671372961254683505791443, 1.92293739266139938144572286158, 1.96399745940612443252913534188, 1.98680973677376612674451772104, 1.99341598981168588209902825551, 2.06163842739989949553834581914, 2.11519299915871821486560519638, 2.16221263729052953047964067842
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.