Dirichlet series
| L(s) = 1 | + 3·2-s + 3·3-s + 3·4-s + 9·6-s − 2·8-s + 6·9-s + 9·12-s + 3·13-s − 12·16-s − 24·17-s + 18·18-s + 9·23-s − 6·24-s + 9·26-s + 8·27-s + 15·29-s − 18·31-s − 24·32-s − 72·34-s + 18·36-s − 36·37-s + 9·39-s − 30·41-s + 36·43-s + 27·46-s − 21·47-s − 36·48-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s + 3.67·6-s − 0.707·8-s + 2·9-s + 2.59·12-s + 0.832·13-s − 3·16-s − 5.82·17-s + 4.24·18-s + 1.87·23-s − 1.22·24-s + 1.76·26-s + 1.53·27-s + 2.78·29-s − 3.23·31-s − 4.24·32-s − 12.3·34-s + 3·36-s − 5.91·37-s + 1.44·39-s − 4.68·41-s + 5.48·43-s + 3.98·46-s − 3.06·47-s − 5.19·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{36} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{36} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(5^{36} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.63920\times 10^{10}\) |
| Root analytic conductor: | \(1.94753\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 5^{36} \cdot 19^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9594823568\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9594823568\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 45 T^{2} + 159 T^{3} + 657 T^{4} + 7137 T^{5} + 17439 T^{6} + 131616 T^{7} + 676053 T^{8} + 2072495 T^{9} + 676053 p T^{10} + 131616 p^{2} T^{11} + 17439 p^{3} T^{12} + 7137 p^{4} T^{13} + 657 p^{5} T^{14} + 159 p^{6} T^{15} + 45 p^{7} T^{16} + p^{9} T^{18} \) | |
| good | 2 | \( 1 - 3 T + 3 p T^{2} - 7 T^{3} + 9 T^{4} - 3 p T^{5} - p^{3} T^{6} + 21 p T^{7} - 21 p^{2} T^{8} + 57 p T^{9} - 21 p^{3} T^{10} + 237 T^{11} - 321 T^{12} + 15 p^{4} T^{13} - 59 p^{3} T^{15} + 723 p T^{16} - 783 p^{2} T^{17} + 5385 T^{18} - 783 p^{3} T^{19} + 723 p^{3} T^{20} - 59 p^{6} T^{21} + 15 p^{9} T^{23} - 321 p^{6} T^{24} + 237 p^{7} T^{25} - 21 p^{11} T^{26} + 57 p^{10} T^{27} - 21 p^{12} T^{28} + 21 p^{12} T^{29} - p^{15} T^{30} - 3 p^{14} T^{31} + 9 p^{14} T^{32} - 7 p^{15} T^{33} + 3 p^{17} T^{34} - 3 p^{17} T^{35} + p^{18} T^{36} \) |
| 3 | \( 1 - p T + p T^{2} + T^{3} - p T^{4} - 10 p T^{5} + 83 T^{6} - 2 p^{2} T^{7} - 83 p T^{8} + 92 p T^{9} + 68 p^{2} T^{10} - 182 p^{2} T^{11} - 26 p^{2} T^{12} + 1952 p T^{13} - 1073 p T^{14} - 19126 T^{15} + 1423 p^{3} T^{16} - 398 p T^{17} - 76769 T^{18} - 398 p^{2} T^{19} + 1423 p^{5} T^{20} - 19126 p^{3} T^{21} - 1073 p^{5} T^{22} + 1952 p^{6} T^{23} - 26 p^{8} T^{24} - 182 p^{9} T^{25} + 68 p^{10} T^{26} + 92 p^{10} T^{27} - 83 p^{11} T^{28} - 2 p^{13} T^{29} + 83 p^{12} T^{30} - 10 p^{14} T^{31} - p^{15} T^{32} + p^{15} T^{33} + p^{17} T^{34} - p^{18} T^{35} + p^{18} T^{36} \) | |
| 7 | \( 1 - 36 T^{2} + 48 T^{3} + 642 T^{4} - 1530 T^{5} - 6554 T^{6} + 3363 p T^{7} + 39489 T^{8} - 208994 T^{9} - 141567 T^{10} + 1028607 T^{11} + 1120991 T^{12} - 856125 T^{13} - 22497312 T^{14} - 22021855 T^{15} + 298649667 T^{16} + 95827002 T^{17} - 2542524043 T^{18} + 95827002 p T^{19} + 298649667 p^{2} T^{20} - 22021855 p^{3} T^{21} - 22497312 p^{4} T^{22} - 856125 p^{5} T^{23} + 1120991 p^{6} T^{24} + 1028607 p^{7} T^{25} - 141567 p^{8} T^{26} - 208994 p^{9} T^{27} + 39489 p^{10} T^{28} + 3363 p^{12} T^{29} - 6554 p^{12} T^{30} - 1530 p^{13} T^{31} + 642 p^{14} T^{32} + 48 p^{15} T^{33} - 36 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( 1 - 39 T^{2} - 104 T^{3} + 567 T^{4} + 3423 T^{5} + 2204 T^{6} - 34350 T^{7} - 11901 p T^{8} - 302264 T^{9} + 77199 T^{10} + 7858746 T^{11} + 37575715 T^{12} + 12298785 T^{13} - 462951186 T^{14} - 1630667133 T^{15} - 523577559 T^{16} + 11350196037 T^{17} + 49238146637 T^{18} + 11350196037 p T^{19} - 523577559 p^{2} T^{20} - 1630667133 p^{3} T^{21} - 462951186 p^{4} T^{22} + 12298785 p^{5} T^{23} + 37575715 p^{6} T^{24} + 7858746 p^{7} T^{25} + 77199 p^{8} T^{26} - 302264 p^{9} T^{27} - 11901 p^{11} T^{28} - 34350 p^{11} T^{29} + 2204 p^{12} T^{30} + 3423 p^{13} T^{31} + 567 p^{14} T^{32} - 104 p^{15} T^{33} - 39 p^{16} T^{34} + p^{18} T^{36} \) | |
| 13 | \( 1 - 3 T + 18 T^{2} + 4 p T^{3} - 219 T^{4} + 2220 T^{5} - 4967 T^{6} + 8418 T^{7} + 63675 T^{8} - 387821 T^{9} + 1297659 T^{10} - 201195 p T^{11} + 164414 T^{12} + 3384294 p T^{13} - 156895236 T^{14} + 657129325 T^{15} + 697310613 T^{16} - 938652978 T^{17} + 31666355345 T^{18} - 938652978 p T^{19} + 697310613 p^{2} T^{20} + 657129325 p^{3} T^{21} - 156895236 p^{4} T^{22} + 3384294 p^{6} T^{23} + 164414 p^{6} T^{24} - 201195 p^{8} T^{25} + 1297659 p^{8} T^{26} - 387821 p^{9} T^{27} + 63675 p^{10} T^{28} + 8418 p^{11} T^{29} - 4967 p^{12} T^{30} + 2220 p^{13} T^{31} - 219 p^{14} T^{32} + 4 p^{16} T^{33} + 18 p^{16} T^{34} - 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( 1 + 24 T + 300 T^{2} + 2440 T^{3} + 14100 T^{4} + 60930 T^{5} + 220651 T^{6} + 912555 T^{7} + 5112273 T^{8} + 27996378 T^{9} + 112809438 T^{10} + 274808112 T^{11} + 200158983 T^{12} - 270173475 T^{13} + 6475889181 T^{14} + 46946480686 T^{15} + 10261792896 T^{16} - 1398528565086 T^{17} - 8796630438597 T^{18} - 1398528565086 p T^{19} + 10261792896 p^{2} T^{20} + 46946480686 p^{3} T^{21} + 6475889181 p^{4} T^{22} - 270173475 p^{5} T^{23} + 200158983 p^{6} T^{24} + 274808112 p^{7} T^{25} + 112809438 p^{8} T^{26} + 27996378 p^{9} T^{27} + 5112273 p^{10} T^{28} + 912555 p^{11} T^{29} + 220651 p^{12} T^{30} + 60930 p^{13} T^{31} + 14100 p^{14} T^{32} + 2440 p^{15} T^{33} + 300 p^{16} T^{34} + 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 - 9 T + 6 T^{2} - 47 T^{3} + 1959 T^{4} - 13281 T^{5} + 62856 T^{6} - 283299 T^{7} + 2152764 T^{8} - 15582803 T^{9} + 69403080 T^{10} - 376009293 T^{11} + 2438556497 T^{12} - 12961683897 T^{13} + 63296680410 T^{14} - 292822536165 T^{15} + 1589247210648 T^{16} - 8473012263072 T^{17} + 39582169481575 T^{18} - 8473012263072 p T^{19} + 1589247210648 p^{2} T^{20} - 292822536165 p^{3} T^{21} + 63296680410 p^{4} T^{22} - 12961683897 p^{5} T^{23} + 2438556497 p^{6} T^{24} - 376009293 p^{7} T^{25} + 69403080 p^{8} T^{26} - 15582803 p^{9} T^{27} + 2152764 p^{10} T^{28} - 283299 p^{11} T^{29} + 62856 p^{12} T^{30} - 13281 p^{13} T^{31} + 1959 p^{14} T^{32} - 47 p^{15} T^{33} + 6 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 - 15 T + 3 p T^{2} - 277 T^{3} + 720 T^{4} + 2820 T^{5} - 30083 T^{6} - 116469 T^{7} + 1713687 T^{8} - 11861364 T^{9} + 86011824 T^{10} - 167800272 T^{11} - 1397578278 T^{12} + 6501167088 T^{13} - 39786701475 T^{14} + 10379907856 p T^{15} - 169213557462 T^{16} - 2583625482843 T^{17} - 1866480481761 T^{18} - 2583625482843 p T^{19} - 169213557462 p^{2} T^{20} + 10379907856 p^{4} T^{21} - 39786701475 p^{4} T^{22} + 6501167088 p^{5} T^{23} - 1397578278 p^{6} T^{24} - 167800272 p^{7} T^{25} + 86011824 p^{8} T^{26} - 11861364 p^{9} T^{27} + 1713687 p^{10} T^{28} - 116469 p^{11} T^{29} - 30083 p^{12} T^{30} + 2820 p^{13} T^{31} + 720 p^{14} T^{32} - 277 p^{15} T^{33} + 3 p^{17} T^{34} - 15 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 + 18 T + 24 T^{2} - 1160 T^{3} - 3246 T^{4} + 37551 T^{5} - 72350 T^{6} - 2269845 T^{7} + 7041792 T^{8} + 104991460 T^{9} - 280009737 T^{10} - 2712457185 T^{11} + 18485068085 T^{12} + 74727912945 T^{13} - 855030421980 T^{14} - 2084260020422 T^{15} + 28537536994635 T^{16} + 25130959002078 T^{17} - 897148690597579 T^{18} + 25130959002078 p T^{19} + 28537536994635 p^{2} T^{20} - 2084260020422 p^{3} T^{21} - 855030421980 p^{4} T^{22} + 74727912945 p^{5} T^{23} + 18485068085 p^{6} T^{24} - 2712457185 p^{7} T^{25} - 280009737 p^{8} T^{26} + 104991460 p^{9} T^{27} + 7041792 p^{10} T^{28} - 2269845 p^{11} T^{29} - 72350 p^{12} T^{30} + 37551 p^{13} T^{31} - 3246 p^{14} T^{32} - 1160 p^{15} T^{33} + 24 p^{16} T^{34} + 18 p^{17} T^{35} + p^{18} T^{36} \) | |
| 37 | \( ( 1 + 18 T + 318 T^{2} + 3704 T^{3} + 39882 T^{4} + 353355 T^{5} + 2925055 T^{6} + 21419565 T^{7} + 3990912 p T^{8} + 925340573 T^{9} + 3990912 p^{2} T^{10} + 21419565 p^{2} T^{11} + 2925055 p^{3} T^{12} + 353355 p^{4} T^{13} + 39882 p^{5} T^{14} + 3704 p^{6} T^{15} + 318 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 + 30 T + 387 T^{2} + 2527 T^{3} + 870 T^{4} - 181677 T^{5} - 2248322 T^{6} - 14126490 T^{7} - 19104291 T^{8} + 639066975 T^{9} + 7766842566 T^{10} + 43608841572 T^{11} + 30284166738 T^{12} - 1883967799110 T^{13} - 19670937570756 T^{14} - 98148403685216 T^{15} - 22014087678252 T^{16} + 4167695527423869 T^{17} + 38741328267440799 T^{18} + 4167695527423869 p T^{19} - 22014087678252 p^{2} T^{20} - 98148403685216 p^{3} T^{21} - 19670937570756 p^{4} T^{22} - 1883967799110 p^{5} T^{23} + 30284166738 p^{6} T^{24} + 43608841572 p^{7} T^{25} + 7766842566 p^{8} T^{26} + 639066975 p^{9} T^{27} - 19104291 p^{10} T^{28} - 14126490 p^{11} T^{29} - 2248322 p^{12} T^{30} - 181677 p^{13} T^{31} + 870 p^{14} T^{32} + 2527 p^{15} T^{33} + 387 p^{16} T^{34} + 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( 1 - 36 T + 696 T^{2} - 9554 T^{3} + 106446 T^{4} - 1034136 T^{5} + 9146626 T^{6} - 75197946 T^{7} + 13518564 p T^{8} - 98309126 p T^{9} + 28742179356 T^{10} - 180662613582 T^{11} + 1040058595520 T^{12} - 125411638338 p T^{13} + 24331364267670 T^{14} - 86772101188100 T^{15} + 166755455447490 T^{16} + 650527611942324 T^{17} - 8177713099035121 T^{18} + 650527611942324 p T^{19} + 166755455447490 p^{2} T^{20} - 86772101188100 p^{3} T^{21} + 24331364267670 p^{4} T^{22} - 125411638338 p^{6} T^{23} + 1040058595520 p^{6} T^{24} - 180662613582 p^{7} T^{25} + 28742179356 p^{8} T^{26} - 98309126 p^{10} T^{27} + 13518564 p^{11} T^{28} - 75197946 p^{11} T^{29} + 9146626 p^{12} T^{30} - 1034136 p^{13} T^{31} + 106446 p^{14} T^{32} - 9554 p^{15} T^{33} + 696 p^{16} T^{34} - 36 p^{17} T^{35} + p^{18} T^{36} \) | |
| 47 | \( 1 + 21 T + 267 T^{2} + 3051 T^{3} + 31053 T^{4} + 310578 T^{5} + 3005629 T^{6} + 568236 p T^{7} + 227057313 T^{8} + 1839795778 T^{9} + 14279996928 T^{10} + 108639838338 T^{11} + 800243198430 T^{12} + 122839761828 p T^{13} + 40768638978129 T^{14} + 280061148901706 T^{15} + 1926488158384497 T^{16} + 13274439448896198 T^{17} + 90946890314655867 T^{18} + 13274439448896198 p T^{19} + 1926488158384497 p^{2} T^{20} + 280061148901706 p^{3} T^{21} + 40768638978129 p^{4} T^{22} + 122839761828 p^{6} T^{23} + 800243198430 p^{6} T^{24} + 108639838338 p^{7} T^{25} + 14279996928 p^{8} T^{26} + 1839795778 p^{9} T^{27} + 227057313 p^{10} T^{28} + 568236 p^{12} T^{29} + 3005629 p^{12} T^{30} + 310578 p^{13} T^{31} + 31053 p^{14} T^{32} + 3051 p^{15} T^{33} + 267 p^{16} T^{34} + 21 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 - 12 T - 105 T^{2} + 1199 T^{3} + 7392 T^{4} - 17097 T^{5} - 547248 T^{6} - 4864947 T^{7} + 36258198 T^{8} + 473955704 T^{9} - 1190040423 T^{10} - 29577627366 T^{11} - 8819139559 T^{12} + 1312072662420 T^{13} + 5915144257734 T^{14} - 36687828641835 T^{15} - 665369435240286 T^{16} + 420559605082254 T^{17} + 45447905627628181 T^{18} + 420559605082254 p T^{19} - 665369435240286 p^{2} T^{20} - 36687828641835 p^{3} T^{21} + 5915144257734 p^{4} T^{22} + 1312072662420 p^{5} T^{23} - 8819139559 p^{6} T^{24} - 29577627366 p^{7} T^{25} - 1190040423 p^{8} T^{26} + 473955704 p^{9} T^{27} + 36258198 p^{10} T^{28} - 4864947 p^{11} T^{29} - 547248 p^{12} T^{30} - 17097 p^{13} T^{31} + 7392 p^{14} T^{32} + 1199 p^{15} T^{33} - 105 p^{16} T^{34} - 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( 1 - 18 T + 189 T^{2} - 2168 T^{3} + 24186 T^{4} - 227727 T^{5} + 2125053 T^{6} - 22519365 T^{7} + 221406408 T^{8} - 1944630014 T^{9} + 17897114184 T^{10} - 161700234537 T^{11} + 1338482359610 T^{12} - 11177711405364 T^{13} + 96161587517781 T^{14} - 787748007704745 T^{15} + 6123255636408297 T^{16} - 48866266581782922 T^{17} + 388910423709609091 T^{18} - 48866266581782922 p T^{19} + 6123255636408297 p^{2} T^{20} - 787748007704745 p^{3} T^{21} + 96161587517781 p^{4} T^{22} - 11177711405364 p^{5} T^{23} + 1338482359610 p^{6} T^{24} - 161700234537 p^{7} T^{25} + 17897114184 p^{8} T^{26} - 1944630014 p^{9} T^{27} + 221406408 p^{10} T^{28} - 22519365 p^{11} T^{29} + 2125053 p^{12} T^{30} - 227727 p^{13} T^{31} + 24186 p^{14} T^{32} - 2168 p^{15} T^{33} + 189 p^{16} T^{34} - 18 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 + 30 T + 342 T^{2} + 2290 T^{3} + 26430 T^{4} + 386583 T^{5} + 3301783 T^{6} + 21197547 T^{7} + 220296195 T^{8} + 2061103690 T^{9} + 8524797915 T^{10} + 29398098612 T^{11} + 573664127609 T^{12} + 2879664163830 T^{13} - 32740735767126 T^{14} - 323902451426225 T^{15} - 928496470308597 T^{16} - 18983934931776267 T^{17} - 267041486596482403 T^{18} - 18983934931776267 p T^{19} - 928496470308597 p^{2} T^{20} - 323902451426225 p^{3} T^{21} - 32740735767126 p^{4} T^{22} + 2879664163830 p^{5} T^{23} + 573664127609 p^{6} T^{24} + 29398098612 p^{7} T^{25} + 8524797915 p^{8} T^{26} + 2061103690 p^{9} T^{27} + 220296195 p^{10} T^{28} + 21197547 p^{11} T^{29} + 3301783 p^{12} T^{30} + 386583 p^{13} T^{31} + 26430 p^{14} T^{32} + 2290 p^{15} T^{33} + 342 p^{16} T^{34} + 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 63 T^{2} + 1711 T^{3} - 300 T^{4} - 27060 T^{5} + 1656290 T^{6} - 3642375 T^{7} + 52445481 T^{8} + 892344245 T^{9} - 4804597863 T^{10} + 99738745731 T^{11} + 231477979532 T^{12} - 1619521139556 T^{13} + 77159173026342 T^{14} - 156589656831939 T^{15} + 1625195377112400 T^{16} + 36690257931319545 T^{17} - 152460929769287427 T^{18} + 36690257931319545 p T^{19} + 1625195377112400 p^{2} T^{20} - 156589656831939 p^{3} T^{21} + 77159173026342 p^{4} T^{22} - 1619521139556 p^{5} T^{23} + 231477979532 p^{6} T^{24} + 99738745731 p^{7} T^{25} - 4804597863 p^{8} T^{26} + 892344245 p^{9} T^{27} + 52445481 p^{10} T^{28} - 3642375 p^{11} T^{29} + 1656290 p^{12} T^{30} - 27060 p^{13} T^{31} - 300 p^{14} T^{32} + 1711 p^{15} T^{33} - 63 p^{16} T^{34} + p^{18} T^{36} \) | |
| 71 | \( 1 + 12 T + 69 T^{2} - 280 T^{3} + 2520 T^{4} + 27078 T^{5} + 243102 T^{6} - 6670329 T^{7} - 41556543 T^{8} - 381059251 T^{9} + 1466622129 T^{10} - 18089038719 T^{11} + 33118448600 T^{12} - 1597464690513 T^{13} + 13226025784425 T^{14} - 13798710826071 T^{15} + 1498386689164614 T^{16} - 71779866183447 p T^{17} + 46874477626963975 T^{18} - 71779866183447 p^{2} T^{19} + 1498386689164614 p^{2} T^{20} - 13798710826071 p^{3} T^{21} + 13226025784425 p^{4} T^{22} - 1597464690513 p^{5} T^{23} + 33118448600 p^{6} T^{24} - 18089038719 p^{7} T^{25} + 1466622129 p^{8} T^{26} - 381059251 p^{9} T^{27} - 41556543 p^{10} T^{28} - 6670329 p^{11} T^{29} + 243102 p^{12} T^{30} + 27078 p^{13} T^{31} + 2520 p^{14} T^{32} - 280 p^{15} T^{33} + 69 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( 1 + 24 T + 381 T^{2} + 5617 T^{3} + 77556 T^{4} + 957624 T^{5} + 11038098 T^{6} + 127278150 T^{7} + 1383039267 T^{8} + 14203803368 T^{9} + 145247448993 T^{10} + 1463995528284 T^{11} + 14191493563493 T^{12} + 134233586011686 T^{13} + 1271197004631270 T^{14} + 11673781441153074 T^{15} + 103864202394246540 T^{16} + 921789455784507207 T^{17} + 8022669989984014591 T^{18} + 921789455784507207 p T^{19} + 103864202394246540 p^{2} T^{20} + 11673781441153074 p^{3} T^{21} + 1271197004631270 p^{4} T^{22} + 134233586011686 p^{5} T^{23} + 14191493563493 p^{6} T^{24} + 1463995528284 p^{7} T^{25} + 145247448993 p^{8} T^{26} + 14203803368 p^{9} T^{27} + 1383039267 p^{10} T^{28} + 127278150 p^{11} T^{29} + 11038098 p^{12} T^{30} + 957624 p^{13} T^{31} + 77556 p^{14} T^{32} + 5617 p^{15} T^{33} + 381 p^{16} T^{34} + 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( 1 + 51 T + 15 p T^{2} + 16260 T^{3} + 151311 T^{4} + 1212447 T^{5} + 10576023 T^{6} + 68835210 T^{7} - 193550889 T^{8} - 11015923757 T^{9} - 148455031401 T^{10} - 1451360213415 T^{11} - 12833471881584 T^{12} - 94601408454270 T^{13} - 412567530589674 T^{14} + 1250515282226139 T^{15} + 51856152307335936 T^{16} + 741510730604472093 T^{17} + 7604193708235168935 T^{18} + 741510730604472093 p T^{19} + 51856152307335936 p^{2} T^{20} + 1250515282226139 p^{3} T^{21} - 412567530589674 p^{4} T^{22} - 94601408454270 p^{5} T^{23} - 12833471881584 p^{6} T^{24} - 1451360213415 p^{7} T^{25} - 148455031401 p^{8} T^{26} - 11015923757 p^{9} T^{27} - 193550889 p^{10} T^{28} + 68835210 p^{11} T^{29} + 10576023 p^{12} T^{30} + 1212447 p^{13} T^{31} + 151311 p^{14} T^{32} + 16260 p^{15} T^{33} + 15 p^{17} T^{34} + 51 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 - 240 T^{2} + 2294 T^{3} + 25227 T^{4} - 573324 T^{5} + 466917 T^{6} + 64782333 T^{7} - 464894352 T^{8} - 3894044566 T^{9} + 64808897223 T^{10} + 35679722685 T^{11} - 5317981062571 T^{12} + 25607882911818 T^{13} + 270977970832401 T^{14} - 3103073741346558 T^{15} + 3306466332395253 T^{16} + 125337633796523766 T^{17} - 1478300132250691877 T^{18} + 125337633796523766 p T^{19} + 3306466332395253 p^{2} T^{20} - 3103073741346558 p^{3} T^{21} + 270977970832401 p^{4} T^{22} + 25607882911818 p^{5} T^{23} - 5317981062571 p^{6} T^{24} + 35679722685 p^{7} T^{25} + 64808897223 p^{8} T^{26} - 3894044566 p^{9} T^{27} - 464894352 p^{10} T^{28} + 64782333 p^{11} T^{29} + 466917 p^{12} T^{30} - 573324 p^{13} T^{31} + 25227 p^{14} T^{32} + 2294 p^{15} T^{33} - 240 p^{16} T^{34} + p^{18} T^{36} \) | |
| 89 | \( 1 + 54 T + 1473 T^{2} + 29411 T^{3} + 498639 T^{4} + 7398054 T^{5} + 97707928 T^{6} + 13239804 p T^{7} + 13014732894 T^{8} + 130934395152 T^{9} + 1199866390443 T^{10} + 9830219834949 T^{11} + 68684711327553 T^{12} + 360387343785321 T^{13} + 408645959323905 T^{14} - 23263622151146428 T^{15} - 423479674715469915 T^{16} - 5268488222149499607 T^{17} - 54147992294647584669 T^{18} - 5268488222149499607 p T^{19} - 423479674715469915 p^{2} T^{20} - 23263622151146428 p^{3} T^{21} + 408645959323905 p^{4} T^{22} + 360387343785321 p^{5} T^{23} + 68684711327553 p^{6} T^{24} + 9830219834949 p^{7} T^{25} + 1199866390443 p^{8} T^{26} + 130934395152 p^{9} T^{27} + 13014732894 p^{10} T^{28} + 13239804 p^{12} T^{29} + 97707928 p^{12} T^{30} + 7398054 p^{13} T^{31} + 498639 p^{14} T^{32} + 29411 p^{15} T^{33} + 1473 p^{16} T^{34} + 54 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( 1 + 27 T + 129 T^{2} - 3203 T^{3} - 40860 T^{4} - 72390 T^{5} + 547648 T^{6} + 8280813 T^{7} + 252510756 T^{8} + 2236148635 T^{9} + 24300311358 T^{10} + 402865541841 T^{11} - 1490704215163 T^{12} - 92108680582047 T^{13} - 519633567183846 T^{14} + 2308545402911368 T^{15} + 21438265253681709 T^{16} + 208828162174255380 T^{17} + 3735862762256338265 T^{18} + 208828162174255380 p T^{19} + 21438265253681709 p^{2} T^{20} + 2308545402911368 p^{3} T^{21} - 519633567183846 p^{4} T^{22} - 92108680582047 p^{5} T^{23} - 1490704215163 p^{6} T^{24} + 402865541841 p^{7} T^{25} + 24300311358 p^{8} T^{26} + 2236148635 p^{9} T^{27} + 252510756 p^{10} T^{28} + 8280813 p^{11} T^{29} + 547648 p^{12} T^{30} - 72390 p^{13} T^{31} - 40860 p^{14} T^{32} - 3203 p^{15} T^{33} + 129 p^{16} T^{34} + 27 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−2.76425044304747317668223343897, −2.74221806925562319355461861248, −2.73695605899648380025597604615, −2.72937188386740448363904859230, −2.61878131100051313217476053359, −2.36470846400093948421602659457, −2.26585644851765820000347716564, −2.24741314047570831038643726168, −2.15611299589223434089874806721, −2.08382692361860356792891476443, −1.95124847651196944955202960595, −1.89833258907225783928805425774, −1.83058125435032782845902047937, −1.71799294448779129338819171518, −1.67337901391075622524720056370, −1.59845897014362309644838130530, −1.58587612879602016515598094043, −1.54839575071696143526044097957, −1.21087242527374100226401911571, −0.990871839795019120223310474888, −0.841655511667984053854185386642, −0.833391530888390168434921862557, −0.48027968962302923575759707462, −0.24544293622944518923946890249, −0.07118817565899405906442109512, 0.07118817565899405906442109512, 0.24544293622944518923946890249, 0.48027968962302923575759707462, 0.833391530888390168434921862557, 0.841655511667984053854185386642, 0.990871839795019120223310474888, 1.21087242527374100226401911571, 1.54839575071696143526044097957, 1.58587612879602016515598094043, 1.59845897014362309644838130530, 1.67337901391075622524720056370, 1.71799294448779129338819171518, 1.83058125435032782845902047937, 1.89833258907225783928805425774, 1.95124847651196944955202960595, 2.08382692361860356792891476443, 2.15611299589223434089874806721, 2.24741314047570831038643726168, 2.26585644851765820000347716564, 2.36470846400093948421602659457, 2.61878131100051313217476053359, 2.72937188386740448363904859230, 2.73695605899648380025597604615, 2.74221806925562319355461861248, 2.76425044304747317668223343897
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.