Dirichlet series
| L(s) = 1 | − 4·2-s − 3·3-s + 8·4-s − 5-s + 12·6-s − 4·7-s − 13·8-s + 3·9-s + 4·10-s + 26·11-s − 24·12-s + 9·13-s + 16·14-s + 3·15-s + 16·16-s + 4·17-s − 12·18-s − 10·19-s − 8·20-s + 12·21-s − 104·22-s − 8·23-s + 39·24-s + 16·25-s − 36·26-s − 27-s − 32·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.73·3-s + 4·4-s − 0.447·5-s + 4.89·6-s − 1.51·7-s − 4.59·8-s + 9-s + 1.26·10-s + 7.83·11-s − 6.92·12-s + 2.49·13-s + 4.27·14-s + 0.774·15-s + 4·16-s + 0.970·17-s − 2.82·18-s − 2.29·19-s − 1.78·20-s + 2.61·21-s − 22.1·22-s − 1.66·23-s + 7.96·24-s + 16/5·25-s − 7.06·26-s − 0.192·27-s − 6.04·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{18} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00142015\) |
| Root analytic conductor: | \(0.833485\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 3^{18} \cdot 29^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06356513193\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06356513193\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3} \) |
| 29 | \( 1 - 8 T + 30 T^{2} - 156 T^{3} + 948 T^{4} + 1579 T^{5} - 13954 T^{6} - 175416 T^{7} + 1130021 T^{8} - 3557529 T^{9} + 1130021 p T^{10} - 175416 p^{2} T^{11} - 13954 p^{3} T^{12} + 1579 p^{4} T^{13} + 948 p^{5} T^{14} - 156 p^{6} T^{15} + 30 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} + 13 T^{3} + 3 p^{3} T^{4} + 41 T^{5} + 31 p T^{6} + 51 p T^{7} + 85 p T^{8} + 119 p T^{9} + 153 p T^{10} + 197 p T^{11} + 249 p T^{12} + 585 T^{13} + 179 p^{2} T^{14} + 835 T^{15} + 923 T^{16} + 531 p T^{17} + 1429 T^{18} + 531 p^{2} T^{19} + 923 p^{2} T^{20} + 835 p^{3} T^{21} + 179 p^{6} T^{22} + 585 p^{5} T^{23} + 249 p^{7} T^{24} + 197 p^{8} T^{25} + 153 p^{9} T^{26} + 119 p^{10} T^{27} + 85 p^{11} T^{28} + 51 p^{12} T^{29} + 31 p^{13} T^{30} + 41 p^{13} T^{31} + 3 p^{17} T^{32} + 13 p^{15} T^{33} + p^{19} T^{34} + p^{19} T^{35} + p^{18} T^{36} \) |
| 5 | \( 1 + T - 3 p T^{2} - 3 T^{3} + 101 T^{4} + 24 T^{5} - 44 T^{6} - 13 p^{2} T^{7} - 2569 T^{8} + 557 p T^{9} + 3532 p T^{10} + 12892 T^{11} - 44583 T^{12} - 8287 p^{2} T^{13} + 42467 T^{14} + 239884 p T^{15} + 1098804 T^{16} - 80079 p^{2} T^{17} - 7048239 T^{18} - 80079 p^{3} T^{19} + 1098804 p^{2} T^{20} + 239884 p^{4} T^{21} + 42467 p^{4} T^{22} - 8287 p^{7} T^{23} - 44583 p^{6} T^{24} + 12892 p^{7} T^{25} + 3532 p^{9} T^{26} + 557 p^{10} T^{27} - 2569 p^{10} T^{28} - 13 p^{13} T^{29} - 44 p^{12} T^{30} + 24 p^{13} T^{31} + 101 p^{14} T^{32} - 3 p^{15} T^{33} - 3 p^{17} T^{34} + p^{17} T^{35} + p^{18} T^{36} \) | |
| 7 | \( 1 + 4 T + 3 p T^{2} + 61 T^{3} + 310 T^{4} + 894 T^{5} + 3455 T^{6} + 1469 p T^{7} + 37444 T^{8} + 15154 p T^{9} + 337048 T^{10} + 974720 T^{11} + 2973617 T^{12} + 8105906 T^{13} + 3444906 p T^{14} + 64299525 T^{15} + 184399825 T^{16} + 463684863 T^{17} + 1318176511 T^{18} + 463684863 p T^{19} + 184399825 p^{2} T^{20} + 64299525 p^{3} T^{21} + 3444906 p^{5} T^{22} + 8105906 p^{5} T^{23} + 2973617 p^{6} T^{24} + 974720 p^{7} T^{25} + 337048 p^{8} T^{26} + 15154 p^{10} T^{27} + 37444 p^{10} T^{28} + 1469 p^{12} T^{29} + 3455 p^{12} T^{30} + 894 p^{13} T^{31} + 310 p^{14} T^{32} + 61 p^{15} T^{33} + 3 p^{17} T^{34} + 4 p^{17} T^{35} + p^{18} T^{36} \) | |
| 11 | \( 1 - 26 T + 302 T^{2} - 2009 T^{3} + 714 p T^{4} - 13722 T^{5} - 22577 T^{6} + 145546 T^{7} + 125210 T^{8} - 2702261 T^{9} + 7201358 T^{10} + 6133988 T^{11} - 81457577 T^{12} + 218351105 T^{13} - 537953352 T^{14} + 2660410686 T^{15} - 882652126 p T^{16} + 13812012211 T^{17} - 1160823201 T^{18} + 13812012211 p T^{19} - 882652126 p^{3} T^{20} + 2660410686 p^{3} T^{21} - 537953352 p^{4} T^{22} + 218351105 p^{5} T^{23} - 81457577 p^{6} T^{24} + 6133988 p^{7} T^{25} + 7201358 p^{8} T^{26} - 2702261 p^{9} T^{27} + 125210 p^{10} T^{28} + 145546 p^{11} T^{29} - 22577 p^{12} T^{30} - 13722 p^{13} T^{31} + 714 p^{15} T^{32} - 2009 p^{15} T^{33} + 302 p^{16} T^{34} - 26 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 - 9 T + 5 T^{2} + 223 T^{3} - 940 T^{4} - 1461 T^{5} + 25645 T^{6} - 55974 T^{7} - 246172 T^{8} + 1357342 T^{9} + 283024 T^{10} - 18679852 T^{11} + 28715569 T^{12} + 190203384 T^{13} - 513041778 T^{14} - 2269013733 T^{15} + 11771865419 T^{16} + 10972150388 T^{17} - 180217490777 T^{18} + 10972150388 p T^{19} + 11771865419 p^{2} T^{20} - 2269013733 p^{3} T^{21} - 513041778 p^{4} T^{22} + 190203384 p^{5} T^{23} + 28715569 p^{6} T^{24} - 18679852 p^{7} T^{25} + 283024 p^{8} T^{26} + 1357342 p^{9} T^{27} - 246172 p^{10} T^{28} - 55974 p^{11} T^{29} + 25645 p^{12} T^{30} - 1461 p^{13} T^{31} - 940 p^{14} T^{32} + 223 p^{15} T^{33} + 5 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( ( 1 - 2 T + 90 T^{2} - 154 T^{3} + 4290 T^{4} - 6733 T^{5} + 135098 T^{6} - 189810 T^{7} + 3077115 T^{8} - 3827129 T^{9} + 3077115 p T^{10} - 189810 p^{2} T^{11} + 135098 p^{3} T^{12} - 6733 p^{4} T^{13} + 4290 p^{5} T^{14} - 154 p^{6} T^{15} + 90 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 19 | \( 1 + 10 T - T^{2} - 191 T^{3} + 243 T^{4} - 1240 T^{5} - 34461 T^{6} + 38438 T^{7} + 635220 T^{8} + 693239 T^{9} + 13608649 T^{10} + 31357596 T^{11} - 15233190 p T^{12} - 185556091 T^{13} - 2507847812 T^{14} - 36214525762 T^{15} + 80460385363 T^{16} + 601195911837 T^{17} - 137819345621 T^{18} + 601195911837 p T^{19} + 80460385363 p^{2} T^{20} - 36214525762 p^{3} T^{21} - 2507847812 p^{4} T^{22} - 185556091 p^{5} T^{23} - 15233190 p^{7} T^{24} + 31357596 p^{7} T^{25} + 13608649 p^{8} T^{26} + 693239 p^{9} T^{27} + 635220 p^{10} T^{28} + 38438 p^{11} T^{29} - 34461 p^{12} T^{30} - 1240 p^{13} T^{31} + 243 p^{14} T^{32} - 191 p^{15} T^{33} - p^{16} T^{34} + 10 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 + 8 T + 17 T^{2} - 186 T^{3} - 864 T^{4} + 2658 T^{5} + 53768 T^{6} + 216136 T^{7} - 42973 T^{8} - 4456868 T^{9} - 9587751 T^{10} + 88765436 T^{11} + 923433288 T^{12} + 3553288252 T^{13} - 1420713443 T^{14} - 75439634080 T^{15} - 164591287208 T^{16} + 1805893573532 T^{17} + 14970559953073 T^{18} + 1805893573532 p T^{19} - 164591287208 p^{2} T^{20} - 75439634080 p^{3} T^{21} - 1420713443 p^{4} T^{22} + 3553288252 p^{5} T^{23} + 923433288 p^{6} T^{24} + 88765436 p^{7} T^{25} - 9587751 p^{8} T^{26} - 4456868 p^{9} T^{27} - 42973 p^{10} T^{28} + 216136 p^{11} T^{29} + 53768 p^{12} T^{30} + 2658 p^{13} T^{31} - 864 p^{14} T^{32} - 186 p^{15} T^{33} + 17 p^{16} T^{34} + 8 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 + 12 T + 76 T^{2} + 87 T^{3} - 89 T^{4} - 4534 T^{5} - 47022 T^{6} - 430437 T^{7} + 1400178 T^{8} + 21739837 T^{9} + 112766587 T^{10} + 138580052 T^{11} + 922495542 T^{12} - 14590863363 T^{13} - 164731648328 T^{14} - 737871956059 T^{15} + 2129310035545 T^{16} + 14836110711401 T^{17} + 88117751323267 T^{18} + 14836110711401 p T^{19} + 2129310035545 p^{2} T^{20} - 737871956059 p^{3} T^{21} - 164731648328 p^{4} T^{22} - 14590863363 p^{5} T^{23} + 922495542 p^{6} T^{24} + 138580052 p^{7} T^{25} + 112766587 p^{8} T^{26} + 21739837 p^{9} T^{27} + 1400178 p^{10} T^{28} - 430437 p^{11} T^{29} - 47022 p^{12} T^{30} - 4534 p^{13} T^{31} - 89 p^{14} T^{32} + 87 p^{15} T^{33} + 76 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 37 | \( 1 + 16 T + 88 T^{2} - 665 T^{3} - 13855 T^{4} - 98021 T^{5} - 127112 T^{6} + 3265931 T^{7} + 35855719 T^{8} + 207171396 T^{9} + 794194458 T^{10} - 818121522 T^{11} - 47576761537 T^{12} - 509277853422 T^{13} - 2846147917167 T^{14} - 6887085960745 T^{15} + 43723012696761 T^{16} + 592710902516744 T^{17} + 4375116437900229 T^{18} + 592710902516744 p T^{19} + 43723012696761 p^{2} T^{20} - 6887085960745 p^{3} T^{21} - 2846147917167 p^{4} T^{22} - 509277853422 p^{5} T^{23} - 47576761537 p^{6} T^{24} - 818121522 p^{7} T^{25} + 794194458 p^{8} T^{26} + 207171396 p^{9} T^{27} + 35855719 p^{10} T^{28} + 3265931 p^{11} T^{29} - 127112 p^{12} T^{30} - 98021 p^{13} T^{31} - 13855 p^{14} T^{32} - 665 p^{15} T^{33} + 88 p^{16} T^{34} + 16 p^{17} T^{35} + p^{18} T^{36} \) | |
| 41 | \( ( 1 - 12 T + 226 T^{2} - 2136 T^{3} + 24552 T^{4} - 199425 T^{5} + 1767092 T^{6} - 12614876 T^{7} + 94224175 T^{8} - 594125521 T^{9} + 94224175 p T^{10} - 12614876 p^{2} T^{11} + 1767092 p^{3} T^{12} - 199425 p^{4} T^{13} + 24552 p^{5} T^{14} - 2136 p^{6} T^{15} + 226 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 43 | \( 1 + 31 T + 275 T^{2} - 1307 T^{3} - 32129 T^{4} + 52337 T^{5} + 3937599 T^{6} + 17569110 T^{7} - 197951339 T^{8} - 1766315555 T^{9} + 8504547470 T^{10} + 156186533546 T^{11} + 162725496837 T^{12} - 7471762958877 T^{13} - 29983957979273 T^{14} + 310239027564311 T^{15} + 2731937459079291 T^{16} - 3947630767518942 T^{17} - 125675025511916367 T^{18} - 3947630767518942 p T^{19} + 2731937459079291 p^{2} T^{20} + 310239027564311 p^{3} T^{21} - 29983957979273 p^{4} T^{22} - 7471762958877 p^{5} T^{23} + 162725496837 p^{6} T^{24} + 156186533546 p^{7} T^{25} + 8504547470 p^{8} T^{26} - 1766315555 p^{9} T^{27} - 197951339 p^{10} T^{28} + 17569110 p^{11} T^{29} + 3937599 p^{12} T^{30} + 52337 p^{13} T^{31} - 32129 p^{14} T^{32} - 1307 p^{15} T^{33} + 275 p^{16} T^{34} + 31 p^{17} T^{35} + p^{18} T^{36} \) | |
| 47 | \( 1 - 5 T - 107 T^{2} - 277 T^{3} + 14880 T^{4} + 16107 T^{5} - 618339 T^{6} - 6040950 T^{7} + 42068648 T^{8} + 229156118 T^{9} + 686907614 T^{10} - 23026548920 T^{11} - 17834179015 T^{12} + 217910890694 T^{13} + 8351686695762 T^{14} - 34066681128107 T^{15} - 166346260303795 T^{16} - 819731160352154 T^{17} + 21765354618363125 T^{18} - 819731160352154 p T^{19} - 166346260303795 p^{2} T^{20} - 34066681128107 p^{3} T^{21} + 8351686695762 p^{4} T^{22} + 217910890694 p^{5} T^{23} - 17834179015 p^{6} T^{24} - 23026548920 p^{7} T^{25} + 686907614 p^{8} T^{26} + 229156118 p^{9} T^{27} + 42068648 p^{10} T^{28} - 6040950 p^{11} T^{29} - 618339 p^{12} T^{30} + 16107 p^{13} T^{31} + 14880 p^{14} T^{32} - 277 p^{15} T^{33} - 107 p^{16} T^{34} - 5 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 - 5 T + 78 T^{2} + 175 T^{3} - 700 T^{4} + 22650 T^{5} + 32387 T^{6} - 1302278 T^{7} - 84979 T^{8} - 56555207 T^{9} - 878259722 T^{10} - 3476468352 T^{11} - 11611376523 T^{12} - 187380096547 T^{13} - 787064453433 T^{14} + 14835561027528 T^{15} + 39916694758707 T^{16} + 206719576053008 T^{17} + 12283781048494768 T^{18} + 206719576053008 p T^{19} + 39916694758707 p^{2} T^{20} + 14835561027528 p^{3} T^{21} - 787064453433 p^{4} T^{22} - 187380096547 p^{5} T^{23} - 11611376523 p^{6} T^{24} - 3476468352 p^{7} T^{25} - 878259722 p^{8} T^{26} - 56555207 p^{9} T^{27} - 84979 p^{10} T^{28} - 1302278 p^{11} T^{29} + 32387 p^{12} T^{30} + 22650 p^{13} T^{31} - 700 p^{14} T^{32} + 175 p^{15} T^{33} + 78 p^{16} T^{34} - 5 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( ( 1 + 16 T + 393 T^{2} + 3630 T^{3} + 49303 T^{4} + 216050 T^{5} + 1990154 T^{6} - 10149150 T^{7} - 52334721 T^{8} - 1597546588 T^{9} - 52334721 p T^{10} - 10149150 p^{2} T^{11} + 1990154 p^{3} T^{12} + 216050 p^{4} T^{13} + 49303 p^{5} T^{14} + 3630 p^{6} T^{15} + 393 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 61 | \( 1 + 28 T + 108 T^{2} - 4645 T^{3} - 54667 T^{4} + 239068 T^{5} + 7699574 T^{6} + 19786749 T^{7} - 572041750 T^{8} - 4200450039 T^{9} + 24996817859 T^{10} + 402207700994 T^{11} - 186445363406 T^{12} - 28883938675493 T^{13} - 103775344816448 T^{14} + 1580086629908023 T^{15} + 14185314020994021 T^{16} - 40309808576648673 T^{17} - 1090223232410592839 T^{18} - 40309808576648673 p T^{19} + 14185314020994021 p^{2} T^{20} + 1580086629908023 p^{3} T^{21} - 103775344816448 p^{4} T^{22} - 28883938675493 p^{5} T^{23} - 186445363406 p^{6} T^{24} + 402207700994 p^{7} T^{25} + 24996817859 p^{8} T^{26} - 4200450039 p^{9} T^{27} - 572041750 p^{10} T^{28} + 19786749 p^{11} T^{29} + 7699574 p^{12} T^{30} + 239068 p^{13} T^{31} - 54667 p^{14} T^{32} - 4645 p^{15} T^{33} + 108 p^{16} T^{34} + 28 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 6 T - 136 T^{2} + 1510 T^{3} + 1728 T^{4} - 154946 T^{5} + 1132355 T^{6} + 2053942 T^{7} - 105767342 T^{8} + 954161648 T^{9} + 207833082 T^{10} - 78446708424 T^{11} + 688481640107 T^{12} - 433348214074 T^{13} - 42665944483610 T^{14} + 387010518285108 T^{15} - 996069316706426 T^{16} - 15968609920338856 T^{17} + 230583104968289273 T^{18} - 15968609920338856 p T^{19} - 996069316706426 p^{2} T^{20} + 387010518285108 p^{3} T^{21} - 42665944483610 p^{4} T^{22} - 433348214074 p^{5} T^{23} + 688481640107 p^{6} T^{24} - 78446708424 p^{7} T^{25} + 207833082 p^{8} T^{26} + 954161648 p^{9} T^{27} - 105767342 p^{10} T^{28} + 2053942 p^{11} T^{29} + 1132355 p^{12} T^{30} - 154946 p^{13} T^{31} + 1728 p^{14} T^{32} + 1510 p^{15} T^{33} - 136 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 - 46 T + 840 T^{2} - 5788 T^{3} - 43588 T^{4} + 1218222 T^{5} - 7671545 T^{6} - 58794642 T^{7} + 1495395114 T^{8} - 10684810134 T^{9} - 24858541540 T^{10} + 1223090764762 T^{11} - 10627375698275 T^{12} + 7765980089046 T^{13} + 768996174786150 T^{14} - 8249143576434214 T^{15} + 22865361147745754 T^{16} + 397421670797090682 T^{17} - 5686465819585021247 T^{18} + 397421670797090682 p T^{19} + 22865361147745754 p^{2} T^{20} - 8249143576434214 p^{3} T^{21} + 768996174786150 p^{4} T^{22} + 7765980089046 p^{5} T^{23} - 10627375698275 p^{6} T^{24} + 1223090764762 p^{7} T^{25} - 24858541540 p^{8} T^{26} - 10684810134 p^{9} T^{27} + 1495395114 p^{10} T^{28} - 58794642 p^{11} T^{29} - 7671545 p^{12} T^{30} + 1218222 p^{13} T^{31} - 43588 p^{14} T^{32} - 5788 p^{15} T^{33} + 840 p^{16} T^{34} - 46 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( 1 + T - 240 T^{2} + 354 T^{3} + 19470 T^{4} - 178177 T^{5} - 120008 T^{6} + 20215298 T^{7} - 103553499 T^{8} - 288715700 T^{9} + 9256987538 T^{10} - 84898770874 T^{11} + 49034626703 T^{12} + 5329148562033 T^{13} - 67888885258176 T^{14} - 1648428159693 p T^{15} + 3542473335105579 T^{16} + 1960710584819638 T^{17} - 52511183002292144 T^{18} + 1960710584819638 p T^{19} + 3542473335105579 p^{2} T^{20} - 1648428159693 p^{4} T^{21} - 67888885258176 p^{4} T^{22} + 5329148562033 p^{5} T^{23} + 49034626703 p^{6} T^{24} - 84898770874 p^{7} T^{25} + 9256987538 p^{8} T^{26} - 288715700 p^{9} T^{27} - 103553499 p^{10} T^{28} + 20215298 p^{11} T^{29} - 120008 p^{12} T^{30} - 178177 p^{13} T^{31} + 19470 p^{14} T^{32} + 354 p^{15} T^{33} - 240 p^{16} T^{34} + p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( 1 + 15 T - 58 T^{2} - 1987 T^{3} + 4389 T^{4} + 321850 T^{5} + 2437768 T^{6} - 11733515 T^{7} - 323519202 T^{8} - 179674507 T^{9} + 41066531577 T^{10} + 314683402260 T^{11} - 1482124097618 T^{12} - 31958542204649 T^{13} - 37368961931708 T^{14} + 2916311522844830 T^{15} + 25893187447010575 T^{16} - 64427079763262691 T^{17} - 2274526317424144057 T^{18} - 64427079763262691 p T^{19} + 25893187447010575 p^{2} T^{20} + 2916311522844830 p^{3} T^{21} - 37368961931708 p^{4} T^{22} - 31958542204649 p^{5} T^{23} - 1482124097618 p^{6} T^{24} + 314683402260 p^{7} T^{25} + 41066531577 p^{8} T^{26} - 179674507 p^{9} T^{27} - 323519202 p^{10} T^{28} - 11733515 p^{11} T^{29} + 2437768 p^{12} T^{30} + 321850 p^{13} T^{31} + 4389 p^{14} T^{32} - 1987 p^{15} T^{33} - 58 p^{16} T^{34} + 15 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 + 16 T - 124 T^{2} - 2538 T^{3} + 8424 T^{4} + 87022 T^{5} - 1236843 T^{6} + 7246096 T^{7} + 78226442 T^{8} - 592989432 T^{9} + 14521662728 T^{10} + 60453078300 T^{11} - 1680315398487 T^{12} + 34874713188 T^{13} - 19067515004594 T^{14} - 1148956207959278 T^{15} + 11718738586618264 T^{16} + 75702500953690466 T^{17} - 1003730765751367715 T^{18} + 75702500953690466 p T^{19} + 11718738586618264 p^{2} T^{20} - 1148956207959278 p^{3} T^{21} - 19067515004594 p^{4} T^{22} + 34874713188 p^{5} T^{23} - 1680315398487 p^{6} T^{24} + 60453078300 p^{7} T^{25} + 14521662728 p^{8} T^{26} - 592989432 p^{9} T^{27} + 78226442 p^{10} T^{28} + 7246096 p^{11} T^{29} - 1236843 p^{12} T^{30} + 87022 p^{13} T^{31} + 8424 p^{14} T^{32} - 2538 p^{15} T^{33} - 124 p^{16} T^{34} + 16 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 + 72 T + 2634 T^{2} + 66950 T^{3} + 1362110 T^{4} + 24016150 T^{5} + 382883333 T^{6} + 5626166368 T^{7} + 76865969274 T^{8} + 984117590046 T^{9} + 11918342986416 T^{10} + 137681581755876 T^{11} + 1525866460996805 T^{12} + 16291954541512044 T^{13} + 168364943659591962 T^{14} + 1693450443028420438 T^{15} + 16667155495954911718 T^{16} + \)\(16\!\cdots\!50\)\( T^{17} + \)\(15\!\cdots\!67\)\( T^{18} + \)\(16\!\cdots\!50\)\( p T^{19} + 16667155495954911718 p^{2} T^{20} + 1693450443028420438 p^{3} T^{21} + 168364943659591962 p^{4} T^{22} + 16291954541512044 p^{5} T^{23} + 1525866460996805 p^{6} T^{24} + 137681581755876 p^{7} T^{25} + 11918342986416 p^{8} T^{26} + 984117590046 p^{9} T^{27} + 76865969274 p^{10} T^{28} + 5626166368 p^{11} T^{29} + 382883333 p^{12} T^{30} + 24016150 p^{13} T^{31} + 1362110 p^{14} T^{32} + 66950 p^{15} T^{33} + 2634 p^{16} T^{34} + 72 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( 1 - 43 T + 961 T^{2} - 16017 T^{3} + 224959 T^{4} - 2697629 T^{5} + 26946586 T^{6} - 221609941 T^{7} + 1501147036 T^{8} - 7449739364 T^{9} + 8616350060 T^{10} + 312667135608 T^{11} - 4988395044509 T^{12} + 74332141235418 T^{13} - 1242769339445498 T^{14} + 19813446871574301 T^{15} - 279267994140950321 T^{16} + 3448000740902678643 T^{17} - 36768886996823203606 T^{18} + 3448000740902678643 p T^{19} - 279267994140950321 p^{2} T^{20} + 19813446871574301 p^{3} T^{21} - 1242769339445498 p^{4} T^{22} + 74332141235418 p^{5} T^{23} - 4988395044509 p^{6} T^{24} + 312667135608 p^{7} T^{25} + 8616350060 p^{8} T^{26} - 7449739364 p^{9} T^{27} + 1501147036 p^{10} T^{28} - 221609941 p^{11} T^{29} + 26946586 p^{12} T^{30} - 2697629 p^{13} T^{31} + 224959 p^{14} T^{32} - 16017 p^{15} T^{33} + 961 p^{16} T^{34} - 43 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−4.30223817013189553533458349916, −3.91975088207769563914483179072, −3.87408855994787623833612340101, −3.87265443812739292850745047022, −3.75731764435968774470892816384, −3.70638328448883270835269901696, −3.66412296839382104671714134181, −3.60024090728967690911123052478, −3.45775692869429023077446155565, −3.33691846232215542560157743956, −3.17500430035784360813652181235, −3.11329208469633293741539300682, −3.09154966484573071098308588733, −2.93429556732389874933229058579, −2.67901717903275616668530623241, −2.54866113577040171331492330518, −2.38999204025373528884186425916, −1.92334256256028243820663637251, −1.91455088683630914974446783842, −1.71581533436500632589262428423, −1.62846935687453105587633689092, −1.59293336240047390911440003872, −1.40518222025409769858113908078, −1.03469816379415901820362250377, −0.938251382334138438353262608340, 0.938251382334138438353262608340, 1.03469816379415901820362250377, 1.40518222025409769858113908078, 1.59293336240047390911440003872, 1.62846935687453105587633689092, 1.71581533436500632589262428423, 1.91455088683630914974446783842, 1.92334256256028243820663637251, 2.38999204025373528884186425916, 2.54866113577040171331492330518, 2.67901717903275616668530623241, 2.93429556732389874933229058579, 3.09154966484573071098308588733, 3.11329208469633293741539300682, 3.17500430035784360813652181235, 3.33691846232215542560157743956, 3.45775692869429023077446155565, 3.60024090728967690911123052478, 3.66412296839382104671714134181, 3.70638328448883270835269901696, 3.75731764435968774470892816384, 3.87265443812739292850745047022, 3.87408855994787623833612340101, 3.91975088207769563914483179072, 4.30223817013189553533458349916
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.