Dirichlet series
| L(s) = 1 | + 6·5-s + 6·11-s − 24·17-s + 3·23-s + 36·25-s + 24·29-s + 33·41-s + 9·47-s + 27·49-s − 66·53-s + 36·55-s + 30·59-s + 9·61-s + 9·67-s − 24·71-s − 18·73-s + 36·83-s − 144·85-s − 96·89-s + 48·101-s + 9·103-s − 30·107-s + 45·113-s + 18·115-s + 63·121-s + 126·125-s + 127-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 1.80·11-s − 5.82·17-s + 0.625·23-s + 36/5·25-s + 4.45·29-s + 5.15·41-s + 1.31·47-s + 27/7·49-s − 9.06·53-s + 4.85·55-s + 3.90·59-s + 1.15·61-s + 1.09·67-s − 2.84·71-s − 2.10·73-s + 3.95·83-s − 15.6·85-s − 10.1·89-s + 4.77·101-s + 0.886·103-s − 2.90·107-s + 4.23·113-s + 1.67·115-s + 5.72·121-s + 11.2·125-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{108}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{108}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{36} \cdot 3^{108}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.04728\times 10^{24}\) |
| Root analytic conductor: | \(4.82538\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{36} \cdot 3^{108} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(37.77442868\) |
| \(L(\frac12)\) | \(\approx\) | \(37.77442868\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 5 | \( 1 - 6 T + 18 p T^{3} - 9 p^{2} T^{4} - 267 T^{5} + 411 p T^{6} - 2499 T^{7} - 3816 T^{8} + 14481 T^{9} - 4977 p T^{10} + 50628 T^{11} - 1716 p T^{12} - 91776 p T^{13} + 1241226 T^{14} - 69642 p T^{15} - 3584223 T^{16} + 5133333 T^{17} - 2059121 T^{18} + 5133333 p T^{19} - 3584223 p^{2} T^{20} - 69642 p^{4} T^{21} + 1241226 p^{4} T^{22} - 91776 p^{6} T^{23} - 1716 p^{7} T^{24} + 50628 p^{7} T^{25} - 4977 p^{9} T^{26} + 14481 p^{9} T^{27} - 3816 p^{10} T^{28} - 2499 p^{11} T^{29} + 411 p^{13} T^{30} - 267 p^{13} T^{31} - 9 p^{16} T^{32} + 18 p^{16} T^{33} - 6 p^{17} T^{35} + p^{18} T^{36} \) |
| 7 | \( 1 - 27 T^{2} - 12 T^{3} + 324 T^{4} + 27 p T^{5} - 2229 T^{6} + 621 T^{7} + 10638 T^{8} - 41260 T^{9} - 52380 T^{10} + 475659 T^{11} + 228780 T^{12} - 2845827 T^{13} + 1699002 T^{14} + 9993096 T^{15} - 48703005 T^{16} - 17505612 T^{17} + 474411093 T^{18} - 17505612 p T^{19} - 48703005 p^{2} T^{20} + 9993096 p^{3} T^{21} + 1699002 p^{4} T^{22} - 2845827 p^{5} T^{23} + 228780 p^{6} T^{24} + 475659 p^{7} T^{25} - 52380 p^{8} T^{26} - 41260 p^{9} T^{27} + 10638 p^{10} T^{28} + 621 p^{11} T^{29} - 2229 p^{12} T^{30} + 27 p^{14} T^{31} + 324 p^{14} T^{32} - 12 p^{15} T^{33} - 27 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( 1 - 6 T - 27 T^{2} + 288 T^{3} - 171 T^{4} - 435 p T^{5} + 12531 T^{6} + 27255 T^{7} - 121644 T^{8} - 2223 T^{9} - 390195 T^{10} + 1927848 T^{11} + 16378476 T^{12} - 58345476 T^{13} - 163151514 T^{14} + 64364364 p T^{15} + 895138020 T^{16} - 3487158537 T^{17} - 3989452979 T^{18} - 3487158537 p T^{19} + 895138020 p^{2} T^{20} + 64364364 p^{4} T^{21} - 163151514 p^{4} T^{22} - 58345476 p^{5} T^{23} + 16378476 p^{6} T^{24} + 1927848 p^{7} T^{25} - 390195 p^{8} T^{26} - 2223 p^{9} T^{27} - 121644 p^{10} T^{28} + 27255 p^{11} T^{29} + 12531 p^{12} T^{30} - 435 p^{14} T^{31} - 171 p^{14} T^{32} + 288 p^{15} T^{33} - 27 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 - 54 T^{2} - 66 T^{3} + 1431 T^{4} + 2673 T^{5} - 21237 T^{6} - 36423 T^{7} + 173880 T^{8} - 248431 T^{9} - 763209 T^{10} + 16656732 T^{11} + 1222596 T^{12} - 271904796 T^{13} + 217927314 T^{14} + 2315030130 T^{15} - 9229435029 T^{16} - 8876738025 T^{17} + 168271978047 T^{18} - 8876738025 p T^{19} - 9229435029 p^{2} T^{20} + 2315030130 p^{3} T^{21} + 217927314 p^{4} T^{22} - 271904796 p^{5} T^{23} + 1222596 p^{6} T^{24} + 16656732 p^{7} T^{25} - 763209 p^{8} T^{26} - 248431 p^{9} T^{27} + 173880 p^{10} T^{28} - 36423 p^{11} T^{29} - 21237 p^{12} T^{30} + 2673 p^{13} T^{31} + 1431 p^{14} T^{32} - 66 p^{15} T^{33} - 54 p^{16} T^{34} + p^{18} T^{36} \) | |
| 17 | \( ( 1 + 12 T + 144 T^{2} + 1179 T^{3} + 9081 T^{4} + 57432 T^{5} + 339261 T^{6} + 1742403 T^{7} + 8380071 T^{8} + 35698797 T^{9} + 8380071 p T^{10} + 1742403 p^{2} T^{11} + 339261 p^{3} T^{12} + 57432 p^{4} T^{13} + 9081 p^{5} T^{14} + 1179 p^{6} T^{15} + 144 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 19 | \( ( 1 + 81 T^{2} - 6 T^{3} + 3645 T^{4} + 108 T^{5} + 6090 p T^{6} + 13149 T^{7} + 2779056 T^{8} + 332377 T^{9} + 2779056 p T^{10} + 13149 p^{2} T^{11} + 6090 p^{4} T^{12} + 108 p^{4} T^{13} + 3645 p^{5} T^{14} - 6 p^{6} T^{15} + 81 p^{7} T^{16} + p^{9} T^{18} )^{2} \) | |
| 23 | \( 1 - 3 T - 81 T^{2} + 144 T^{3} + 4149 T^{4} - 2442 T^{5} - 7257 p T^{6} - 80076 T^{7} + 5313915 T^{8} + 6853635 T^{9} - 131439852 T^{10} - 280270191 T^{11} + 2644315962 T^{12} + 7942872723 T^{13} - 43468870176 T^{14} - 144816041988 T^{15} + 579618022644 T^{16} + 1216904791794 T^{17} - 9363488992031 T^{18} + 1216904791794 p T^{19} + 579618022644 p^{2} T^{20} - 144816041988 p^{3} T^{21} - 43468870176 p^{4} T^{22} + 7942872723 p^{5} T^{23} + 2644315962 p^{6} T^{24} - 280270191 p^{7} T^{25} - 131439852 p^{8} T^{26} + 6853635 p^{9} T^{27} + 5313915 p^{10} T^{28} - 80076 p^{11} T^{29} - 7257 p^{13} T^{30} - 2442 p^{13} T^{31} + 4149 p^{14} T^{32} + 144 p^{15} T^{33} - 81 p^{16} T^{34} - 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 - 24 T + 189 T^{2} - 54 T^{3} - 7164 T^{4} + 31728 T^{5} - 2022 T^{6} + 320646 T^{7} - 4919760 T^{8} - 4893156 T^{9} + 236675826 T^{10} - 26756742 p T^{11} - 1278846705 T^{12} + 48357942 p T^{13} + 53667070758 T^{14} + 311328901836 T^{15} - 3967453770993 T^{16} - 2096609963802 T^{17} + 119684775963841 T^{18} - 2096609963802 p T^{19} - 3967453770993 p^{2} T^{20} + 311328901836 p^{3} T^{21} + 53667070758 p^{4} T^{22} + 48357942 p^{6} T^{23} - 1278846705 p^{6} T^{24} - 26756742 p^{8} T^{25} + 236675826 p^{8} T^{26} - 4893156 p^{9} T^{27} - 4919760 p^{10} T^{28} + 320646 p^{11} T^{29} - 2022 p^{12} T^{30} + 31728 p^{13} T^{31} - 7164 p^{14} T^{32} - 54 p^{15} T^{33} + 189 p^{16} T^{34} - 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 - 135 T^{2} - 336 T^{3} + 9666 T^{4} + 47277 T^{5} - 414519 T^{6} - 3599505 T^{7} + 9028908 T^{8} + 182782544 T^{9} + 4847364 p T^{10} - 6728231223 T^{11} - 23060072322 T^{12} + 183729940389 T^{13} + 1192785320862 T^{14} - 3616163856288 T^{15} - 45004299054543 T^{16} + 37192726239912 T^{17} + 1462295587048665 T^{18} + 37192726239912 p T^{19} - 45004299054543 p^{2} T^{20} - 3616163856288 p^{3} T^{21} + 1192785320862 p^{4} T^{22} + 183729940389 p^{5} T^{23} - 23060072322 p^{6} T^{24} - 6728231223 p^{7} T^{25} + 4847364 p^{9} T^{26} + 182782544 p^{9} T^{27} + 9028908 p^{10} T^{28} - 3599505 p^{11} T^{29} - 414519 p^{12} T^{30} + 47277 p^{13} T^{31} + 9666 p^{14} T^{32} - 336 p^{15} T^{33} - 135 p^{16} T^{34} + p^{18} T^{36} \) | |
| 37 | \( ( 1 + 162 T^{2} - 249 T^{3} + 14175 T^{4} - 28782 T^{5} + 898845 T^{6} - 1802115 T^{7} + 42836499 T^{8} - 79679531 T^{9} + 42836499 p T^{10} - 1802115 p^{2} T^{11} + 898845 p^{3} T^{12} - 28782 p^{4} T^{13} + 14175 p^{5} T^{14} - 249 p^{6} T^{15} + 162 p^{7} T^{16} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 - 33 T + 405 T^{2} - 1908 T^{3} + 612 T^{4} - 29508 T^{5} + 851610 T^{6} - 7408275 T^{7} + 42262659 T^{8} - 110265939 T^{9} - 489750588 T^{10} - 3694127550 T^{11} + 123982409895 T^{12} - 793026997035 T^{13} + 1321605235737 T^{14} + 8788413723858 T^{15} - 31528936461690 T^{16} - 781368850226868 T^{17} + 9211612539491683 T^{18} - 781368850226868 p T^{19} - 31528936461690 p^{2} T^{20} + 8788413723858 p^{3} T^{21} + 1321605235737 p^{4} T^{22} - 793026997035 p^{5} T^{23} + 123982409895 p^{6} T^{24} - 3694127550 p^{7} T^{25} - 489750588 p^{8} T^{26} - 110265939 p^{9} T^{27} + 42262659 p^{10} T^{28} - 7408275 p^{11} T^{29} + 851610 p^{12} T^{30} - 29508 p^{13} T^{31} + 612 p^{14} T^{32} - 1908 p^{15} T^{33} + 405 p^{16} T^{34} - 33 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( 1 - 216 T^{2} - 336 T^{3} + 21411 T^{4} + 61857 T^{5} - 1442274 T^{6} - 2538 p^{2} T^{7} + 88709337 T^{8} + 209542055 T^{9} - 5543975502 T^{10} - 8533455867 T^{11} + 321238098621 T^{12} + 406259307162 T^{13} - 16414550810478 T^{14} - 15284854838772 T^{15} + 761426963627733 T^{16} + 258344045655129 T^{17} - 33398977853201475 T^{18} + 258344045655129 p T^{19} + 761426963627733 p^{2} T^{20} - 15284854838772 p^{3} T^{21} - 16414550810478 p^{4} T^{22} + 406259307162 p^{5} T^{23} + 321238098621 p^{6} T^{24} - 8533455867 p^{7} T^{25} - 5543975502 p^{8} T^{26} + 209542055 p^{9} T^{27} + 88709337 p^{10} T^{28} - 2538 p^{13} T^{29} - 1442274 p^{12} T^{30} + 61857 p^{13} T^{31} + 21411 p^{14} T^{32} - 336 p^{15} T^{33} - 216 p^{16} T^{34} + p^{18} T^{36} \) | |
| 47 | \( 1 - 9 T - 126 T^{2} + 1953 T^{3} - 1251 T^{4} - 127800 T^{5} + 772302 T^{6} + 1073322 T^{7} - 21199608 T^{8} + 90972432 T^{9} - 1150696458 T^{10} + 5827656744 T^{11} + 33770891859 T^{12} - 289258871130 T^{13} + 419039780121 T^{14} - 18549034293609 T^{15} + 198257235556608 T^{16} + 818368073896725 T^{17} - 18162035932650275 T^{18} + 818368073896725 p T^{19} + 198257235556608 p^{2} T^{20} - 18549034293609 p^{3} T^{21} + 419039780121 p^{4} T^{22} - 289258871130 p^{5} T^{23} + 33770891859 p^{6} T^{24} + 5827656744 p^{7} T^{25} - 1150696458 p^{8} T^{26} + 90972432 p^{9} T^{27} - 21199608 p^{10} T^{28} + 1073322 p^{11} T^{29} + 772302 p^{12} T^{30} - 127800 p^{13} T^{31} - 1251 p^{14} T^{32} + 1953 p^{15} T^{33} - 126 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( ( 1 + 33 T + 711 T^{2} + 11529 T^{3} + 156285 T^{4} + 1818492 T^{5} + 18782340 T^{6} + 173567805 T^{7} + 1455502203 T^{8} + 11084684058 T^{9} + 1455502203 p T^{10} + 173567805 p^{2} T^{11} + 18782340 p^{3} T^{12} + 1818492 p^{4} T^{13} + 156285 p^{5} T^{14} + 11529 p^{6} T^{15} + 711 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 59 | \( 1 - 30 T + 243 T^{2} - 450 T^{3} + 30096 T^{4} - 563421 T^{5} + 2682291 T^{6} - 17230395 T^{7} + 452620242 T^{8} - 3514527594 T^{9} + 15216552468 T^{10} - 282765244503 T^{11} + 2724245707110 T^{12} - 10672151656017 T^{13} + 136951045880400 T^{14} - 1622891396671812 T^{15} + 7431947841925737 T^{16} - 61951348685572752 T^{17} + 748816392486302773 T^{18} - 61951348685572752 p T^{19} + 7431947841925737 p^{2} T^{20} - 1622891396671812 p^{3} T^{21} + 136951045880400 p^{4} T^{22} - 10672151656017 p^{5} T^{23} + 2724245707110 p^{6} T^{24} - 282765244503 p^{7} T^{25} + 15216552468 p^{8} T^{26} - 3514527594 p^{9} T^{27} + 452620242 p^{10} T^{28} - 17230395 p^{11} T^{29} + 2682291 p^{12} T^{30} - 563421 p^{13} T^{31} + 30096 p^{14} T^{32} - 450 p^{15} T^{33} + 243 p^{16} T^{34} - 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 - 9 T - 342 T^{2} + 2877 T^{3} + 64521 T^{4} - 484722 T^{5} - 8863224 T^{6} + 58129614 T^{7} + 973764216 T^{8} - 5439121816 T^{9} - 90614020302 T^{10} + 408526596684 T^{11} + 7439991002379 T^{12} - 24551675294778 T^{13} - 552124403027487 T^{14} + 1101212394381387 T^{15} + 37624449982601682 T^{16} - 24679143197070615 T^{17} - 2378821844030380563 T^{18} - 24679143197070615 p T^{19} + 37624449982601682 p^{2} T^{20} + 1101212394381387 p^{3} T^{21} - 552124403027487 p^{4} T^{22} - 24551675294778 p^{5} T^{23} + 7439991002379 p^{6} T^{24} + 408526596684 p^{7} T^{25} - 90614020302 p^{8} T^{26} - 5439121816 p^{9} T^{27} + 973764216 p^{10} T^{28} + 58129614 p^{11} T^{29} - 8863224 p^{12} T^{30} - 484722 p^{13} T^{31} + 64521 p^{14} T^{32} + 2877 p^{15} T^{33} - 342 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 9 T - 369 T^{2} + 2634 T^{3} + 82044 T^{4} - 437418 T^{5} - 12470856 T^{6} + 44572293 T^{7} + 1423554975 T^{8} - 2858963557 T^{9} - 122248930104 T^{10} + 62853352002 T^{11} + 7843304958381 T^{12} + 6850192152393 T^{13} - 342276593924133 T^{14} - 805058836134228 T^{15} + 8128343405945628 T^{16} + 25390602354058812 T^{17} - 85047957679245735 T^{18} + 25390602354058812 p T^{19} + 8128343405945628 p^{2} T^{20} - 805058836134228 p^{3} T^{21} - 342276593924133 p^{4} T^{22} + 6850192152393 p^{5} T^{23} + 7843304958381 p^{6} T^{24} + 62853352002 p^{7} T^{25} - 122248930104 p^{8} T^{26} - 2858963557 p^{9} T^{27} + 1423554975 p^{10} T^{28} + 44572293 p^{11} T^{29} - 12470856 p^{12} T^{30} - 437418 p^{13} T^{31} + 82044 p^{14} T^{32} + 2634 p^{15} T^{33} - 369 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( ( 1 + 12 T + 360 T^{2} + 4149 T^{3} + 70911 T^{4} + 737616 T^{5} + 9482055 T^{6} + 85915713 T^{7} + 914603427 T^{8} + 7123440231 T^{9} + 914603427 p T^{10} + 85915713 p^{2} T^{11} + 9482055 p^{3} T^{12} + 737616 p^{4} T^{13} + 70911 p^{5} T^{14} + 4149 p^{6} T^{15} + 360 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 73 | \( ( 1 + 9 T + 504 T^{2} + 3855 T^{3} + 115083 T^{4} + 762471 T^{5} + 16117503 T^{6} + 93606903 T^{7} + 1579876641 T^{8} + 8028229525 T^{9} + 1579876641 p T^{10} + 93606903 p^{2} T^{11} + 16117503 p^{3} T^{12} + 762471 p^{4} T^{13} + 115083 p^{5} T^{14} + 3855 p^{6} T^{15} + 504 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 79 | \( 1 - 513 T^{2} - 282 T^{3} + 135891 T^{4} + 120933 T^{5} - 25262349 T^{6} - 24043473 T^{7} + 3739433742 T^{8} + 3033149027 T^{9} - 470645380503 T^{10} - 276432567210 T^{11} + 52216811242956 T^{12} + 19140766980282 T^{13} - 5205944985456402 T^{14} - 981014808532914 T^{15} + 471228813588439242 T^{16} + 26206292765214549 T^{17} - 38933236527328138791 T^{18} + 26206292765214549 p T^{19} + 471228813588439242 p^{2} T^{20} - 981014808532914 p^{3} T^{21} - 5205944985456402 p^{4} T^{22} + 19140766980282 p^{5} T^{23} + 52216811242956 p^{6} T^{24} - 276432567210 p^{7} T^{25} - 470645380503 p^{8} T^{26} + 3033149027 p^{9} T^{27} + 3739433742 p^{10} T^{28} - 24043473 p^{11} T^{29} - 25262349 p^{12} T^{30} + 120933 p^{13} T^{31} + 135891 p^{14} T^{32} - 282 p^{15} T^{33} - 513 p^{16} T^{34} + p^{18} T^{36} \) | |
| 83 | \( 1 - 36 T + 333 T^{2} - 504 T^{3} + 62118 T^{4} - 1247733 T^{5} + 3676233 T^{6} - 36347085 T^{7} + 1910669400 T^{8} - 12163787406 T^{9} + 15316111008 T^{10} - 23217561675 p T^{11} + 19580949347388 T^{12} - 15096552025725 T^{13} + 1483485903129654 T^{14} - 22992490501460334 T^{15} + 31725779003589591 T^{16} - 747512729893194660 T^{17} + 20397822173258773237 T^{18} - 747512729893194660 p T^{19} + 31725779003589591 p^{2} T^{20} - 22992490501460334 p^{3} T^{21} + 1483485903129654 p^{4} T^{22} - 15096552025725 p^{5} T^{23} + 19580949347388 p^{6} T^{24} - 23217561675 p^{8} T^{25} + 15316111008 p^{8} T^{26} - 12163787406 p^{9} T^{27} + 1910669400 p^{10} T^{28} - 36347085 p^{11} T^{29} + 3676233 p^{12} T^{30} - 1247733 p^{13} T^{31} + 62118 p^{14} T^{32} - 504 p^{15} T^{33} + 333 p^{16} T^{34} - 36 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( ( 1 + 48 T + 1575 T^{2} + 37530 T^{3} + 739701 T^{4} + 12211194 T^{5} + 175914996 T^{6} + 2216438067 T^{7} + 24879684762 T^{8} + 247959184365 T^{9} + 24879684762 p T^{10} + 2216438067 p^{2} T^{11} + 175914996 p^{3} T^{12} + 12211194 p^{4} T^{13} + 739701 p^{5} T^{14} + 37530 p^{6} T^{15} + 1575 p^{7} T^{16} + 48 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 97 | \( 1 - 594 T^{2} - 876 T^{3} + 184275 T^{4} + 483381 T^{5} - 38984208 T^{6} - 141528708 T^{7} + 6274453401 T^{8} + 27991883471 T^{9} - 811718715294 T^{10} - 4077039967425 T^{11} + 87847144833285 T^{12} + 445752546962508 T^{13} - 8321985929683800 T^{14} - 34602541931613582 T^{15} + 742319689365300771 T^{16} + 1285842891402436113 T^{17} - 68775228635407804101 T^{18} + 1285842891402436113 p T^{19} + 742319689365300771 p^{2} T^{20} - 34602541931613582 p^{3} T^{21} - 8321985929683800 p^{4} T^{22} + 445752546962508 p^{5} T^{23} + 87847144833285 p^{6} T^{24} - 4077039967425 p^{7} T^{25} - 811718715294 p^{8} T^{26} + 27991883471 p^{9} T^{27} + 6274453401 p^{10} T^{28} - 141528708 p^{11} T^{29} - 38984208 p^{12} T^{30} + 483381 p^{13} T^{31} + 184275 p^{14} T^{32} - 876 p^{15} T^{33} - 594 p^{16} T^{34} + p^{18} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.00922870027322957710427922092, −1.98388369243327353433441415920, −1.83830006989230140052673363636, −1.69041378958462286813919965594, −1.67058406914413028084116112874, −1.60663973944207372004622465589, −1.55520362052061587040261942602, −1.48142704440398285843650212646, −1.44880078098479463988045791414, −1.37335898953251059465771946976, −1.29153531960788925100432093448, −1.16214951711546275237027577479, −1.11032423137268259676961672680, −1.10061853454478090165206407923, −1.08449860181659386307564143721, −0.961702611071096882140951055600, −0.815493494453872544550225776015, −0.798390993231997104520702312397, −0.70861643848905351228241906537, −0.58170567922907997962137699418, −0.49911024912817588582850663668, −0.43554543400391074563327197314, −0.42750323400875654803232977240, −0.25698974499101017798682781234, −0.06682513209066703819295096524, 0.06682513209066703819295096524, 0.25698974499101017798682781234, 0.42750323400875654803232977240, 0.43554543400391074563327197314, 0.49911024912817588582850663668, 0.58170567922907997962137699418, 0.70861643848905351228241906537, 0.798390993231997104520702312397, 0.815493494453872544550225776015, 0.961702611071096882140951055600, 1.08449860181659386307564143721, 1.10061853454478090165206407923, 1.11032423137268259676961672680, 1.16214951711546275237027577479, 1.29153531960788925100432093448, 1.37335898953251059465771946976, 1.44880078098479463988045791414, 1.48142704440398285843650212646, 1.55520362052061587040261942602, 1.60663973944207372004622465589, 1.67058406914413028084116112874, 1.69041378958462286813919965594, 1.83830006989230140052673363636, 1.98388369243327353433441415920, 2.00922870027322957710427922092
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.