Dirichlet series
| L(s) = 1 | + 4·2-s − 3·3-s + 15·4-s − 3·5-s − 12·6-s − 3·7-s + 40·8-s + 4·9-s − 12·10-s + 2·11-s − 45·12-s + 27·13-s − 12·14-s + 9·15-s + 103·16-s + 16·17-s + 16·18-s − 4·19-s − 45·20-s + 9·21-s + 8·22-s − 21·23-s − 120·24-s + 18·25-s + 108·26-s + 3·27-s − 45·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.73·3-s + 15/2·4-s − 1.34·5-s − 4.89·6-s − 1.13·7-s + 14.1·8-s + 4/3·9-s − 3.79·10-s + 0.603·11-s − 12.9·12-s + 7.48·13-s − 3.20·14-s + 2.32·15-s + 25.7·16-s + 3.88·17-s + 3.77·18-s − 0.917·19-s − 10.0·20-s + 1.96·21-s + 1.70·22-s − 4.37·23-s − 24.4·24-s + 18/5·25-s + 21.1·26-s + 0.577·27-s − 8.50·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(31^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44546\times 10^{14}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 31^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02795369624\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02795369624\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + T^{2} + p^{4} T^{3} - 11 p T^{4} - 3 p T^{5} + 41 T^{6} - 37 p T^{7} + 27 p T^{8} + 9 p^{4} T^{9} - 317 T^{10} + 59 p T^{11} + 275 T^{12} - 733 T^{13} + 915 T^{14} + 697 T^{15} - 3065 T^{16} + 697 p T^{17} + 915 p^{2} T^{18} - 733 p^{3} T^{19} + 275 p^{4} T^{20} + 59 p^{6} T^{21} - 317 p^{6} T^{22} + 9 p^{11} T^{23} + 27 p^{9} T^{24} - 37 p^{10} T^{25} + 41 p^{10} T^{26} - 3 p^{12} T^{27} - 11 p^{13} T^{28} + p^{17} T^{29} + p^{14} T^{30} - p^{17} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + p T + 5 T^{2} - 2 p^{2} T^{4} - 7 p^{2} T^{5} - 80 T^{6} + 13 p T^{7} + 410 T^{8} + 260 p T^{9} + 637 T^{10} - 55 p^{3} T^{11} - 1928 p T^{12} - 3158 p T^{13} - 1420 T^{14} + 7438 p T^{15} + 59593 T^{16} + 7438 p^{2} T^{17} - 1420 p^{2} T^{18} - 3158 p^{4} T^{19} - 1928 p^{5} T^{20} - 55 p^{8} T^{21} + 637 p^{6} T^{22} + 260 p^{8} T^{23} + 410 p^{8} T^{24} + 13 p^{10} T^{25} - 80 p^{10} T^{26} - 7 p^{13} T^{27} - 2 p^{14} T^{28} + 5 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 + 3 T - 9 T^{2} - 18 T^{3} + 61 T^{4} - 21 T^{5} - 291 T^{6} + 591 T^{7} + 172 T^{8} - 4164 T^{9} + 1989 T^{10} + 14124 T^{11} - 21959 T^{12} - 41793 T^{13} + 231951 T^{14} + 136788 T^{15} - 1318469 T^{16} + 136788 p T^{17} + 231951 p^{2} T^{18} - 41793 p^{3} T^{19} - 21959 p^{4} T^{20} + 14124 p^{5} T^{21} + 1989 p^{6} T^{22} - 4164 p^{7} T^{23} + 172 p^{8} T^{24} + 591 p^{9} T^{25} - 291 p^{10} T^{26} - 21 p^{11} T^{27} + 61 p^{12} T^{28} - 18 p^{13} T^{29} - 9 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 3 T - 2 T^{2} + 4 p T^{3} + 135 T^{4} + 53 T^{5} + 492 T^{6} + 1546 T^{7} + 223 T^{8} + 13583 T^{9} + 14101 T^{10} - 67978 T^{11} + 5448 T^{12} - 165699 T^{13} - 980866 T^{14} - 3114966 T^{15} - 14726113 T^{16} - 3114966 p T^{17} - 980866 p^{2} T^{18} - 165699 p^{3} T^{19} + 5448 p^{4} T^{20} - 67978 p^{5} T^{21} + 14101 p^{6} T^{22} + 13583 p^{7} T^{23} + 223 p^{8} T^{24} + 1546 p^{9} T^{25} + 492 p^{10} T^{26} + 53 p^{11} T^{27} + 135 p^{12} T^{28} + 4 p^{14} T^{29} - 2 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 2 T + 20 T^{2} - 133 T^{3} + 383 T^{4} - 168 p T^{5} + 8080 T^{6} - 20 p^{3} T^{7} + 72789 T^{8} - 346557 T^{9} + 1062359 T^{10} - 2133385 T^{11} + 918129 p T^{12} - 30908828 T^{13} + 48698451 T^{14} - 173439997 T^{15} + 916310367 T^{16} - 173439997 p T^{17} + 48698451 p^{2} T^{18} - 30908828 p^{3} T^{19} + 918129 p^{5} T^{20} - 2133385 p^{5} T^{21} + 1062359 p^{6} T^{22} - 346557 p^{7} T^{23} + 72789 p^{8} T^{24} - 20 p^{12} T^{25} + 8080 p^{10} T^{26} - 168 p^{12} T^{27} + 383 p^{12} T^{28} - 133 p^{13} T^{29} + 20 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 27 T + 379 T^{2} - 283 p T^{3} + 2130 p T^{4} - 171365 T^{5} + 902700 T^{6} - 4132666 T^{7} + 16625836 T^{8} - 58906271 T^{9} + 182243590 T^{10} - 478268177 T^{11} + 974243463 T^{12} - 972425694 T^{13} - 3617324668 T^{14} + 28834808781 T^{15} - 122944078429 T^{16} + 28834808781 p T^{17} - 3617324668 p^{2} T^{18} - 972425694 p^{3} T^{19} + 974243463 p^{4} T^{20} - 478268177 p^{5} T^{21} + 182243590 p^{6} T^{22} - 58906271 p^{7} T^{23} + 16625836 p^{8} T^{24} - 4132666 p^{9} T^{25} + 902700 p^{10} T^{26} - 171365 p^{11} T^{27} + 2130 p^{13} T^{28} - 283 p^{14} T^{29} + 379 p^{14} T^{30} - 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 16 T + 145 T^{2} - 905 T^{3} + 3692 T^{4} - 7254 T^{5} - 28780 T^{6} + 325168 T^{7} - 1370505 T^{8} + 1560480 T^{9} + 21568768 T^{10} - 183966245 T^{11} + 848136476 T^{12} - 2336699212 T^{13} + 985693815 T^{14} + 28296805588 T^{15} - 171509387807 T^{16} + 28296805588 p T^{17} + 985693815 p^{2} T^{18} - 2336699212 p^{3} T^{19} + 848136476 p^{4} T^{20} - 183966245 p^{5} T^{21} + 21568768 p^{6} T^{22} + 1560480 p^{7} T^{23} - 1370505 p^{8} T^{24} + 325168 p^{9} T^{25} - 28780 p^{10} T^{26} - 7254 p^{11} T^{27} + 3692 p^{12} T^{28} - 905 p^{13} T^{29} + 145 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 4 T + 25 T^{2} + 204 T^{3} - 297 T^{4} - 141 p T^{5} - 22110 T^{6} - 204825 T^{7} - 329991 T^{8} - 427371 T^{9} + 497529 p T^{10} + 103717350 T^{11} + 283431099 T^{12} + 1199684976 T^{13} - 1665098811 T^{14} - 37205201459 T^{15} - 121289477783 T^{16} - 37205201459 p T^{17} - 1665098811 p^{2} T^{18} + 1199684976 p^{3} T^{19} + 283431099 p^{4} T^{20} + 103717350 p^{5} T^{21} + 497529 p^{7} T^{22} - 427371 p^{7} T^{23} - 329991 p^{8} T^{24} - 204825 p^{9} T^{25} - 22110 p^{10} T^{26} - 141 p^{12} T^{27} - 297 p^{12} T^{28} + 204 p^{13} T^{29} + 25 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 21 T + 123 T^{2} - 585 T^{3} - 10795 T^{4} - 41532 T^{5} + 99933 T^{6} + 1538064 T^{7} + 7137300 T^{8} + 17794740 T^{9} - 55986288 T^{10} - 980369838 T^{11} - 4973180624 T^{12} - 9063276630 T^{13} + 32490247020 T^{14} + 362608817964 T^{15} + 2051124895679 T^{16} + 362608817964 p T^{17} + 32490247020 p^{2} T^{18} - 9063276630 p^{3} T^{19} - 4973180624 p^{4} T^{20} - 980369838 p^{5} T^{21} - 55986288 p^{6} T^{22} + 17794740 p^{7} T^{23} + 7137300 p^{8} T^{24} + 1538064 p^{9} T^{25} + 99933 p^{10} T^{26} - 41532 p^{11} T^{27} - 10795 p^{12} T^{28} - 585 p^{13} T^{29} + 123 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 26 T + 215 T^{2} - 86 T^{3} - 10816 T^{4} - 35736 T^{5} + 10076 p T^{6} + 1947748 T^{7} - 5844798 T^{8} - 86913771 T^{9} - 60844645 T^{10} + 2748208849 T^{11} + 10600412262 T^{12} - 52411368583 T^{13} - 15641492601 p T^{14} + 293962662715 T^{15} + 12257987383833 T^{16} + 293962662715 p T^{17} - 15641492601 p^{3} T^{18} - 52411368583 p^{3} T^{19} + 10600412262 p^{4} T^{20} + 2748208849 p^{5} T^{21} - 60844645 p^{6} T^{22} - 86913771 p^{7} T^{23} - 5844798 p^{8} T^{24} + 1947748 p^{9} T^{25} + 10076 p^{11} T^{26} - 35736 p^{11} T^{27} - 10816 p^{12} T^{28} - 86 p^{13} T^{29} + 215 p^{14} T^{30} + 26 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 8 T - 120 T^{2} + 534 T^{3} + 11739 T^{4} - 10042 T^{5} - 710515 T^{6} - 2077152 T^{7} + 30531852 T^{8} + 207297908 T^{9} - 600747795 T^{10} - 11697189185 T^{11} - 16371022929 T^{12} + 395056153140 T^{13} + 2176783134259 T^{14} - 6190914463491 T^{15} - 102798721686295 T^{16} - 6190914463491 p T^{17} + 2176783134259 p^{2} T^{18} + 395056153140 p^{3} T^{19} - 16371022929 p^{4} T^{20} - 11697189185 p^{5} T^{21} - 600747795 p^{6} T^{22} + 207297908 p^{7} T^{23} + 30531852 p^{8} T^{24} - 2077152 p^{9} T^{25} - 710515 p^{10} T^{26} - 10042 p^{11} T^{27} + 11739 p^{12} T^{28} + 534 p^{13} T^{29} - 120 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 3 T + 51 T^{2} + 441 T^{3} + 1100 T^{4} + 7695 T^{5} - 1176 T^{6} - 75228 T^{7} - 3079179 T^{8} - 12785988 T^{9} - 122844942 T^{10} - 694048725 T^{11} - 933212414 T^{12} - 8533923600 T^{13} + 282558978282 T^{14} + 2329689004992 T^{15} + 13538379049295 T^{16} + 2329689004992 p T^{17} + 282558978282 p^{2} T^{18} - 8533923600 p^{3} T^{19} - 933212414 p^{4} T^{20} - 694048725 p^{5} T^{21} - 122844942 p^{6} T^{22} - 12785988 p^{7} T^{23} - 3079179 p^{8} T^{24} - 75228 p^{9} T^{25} - 1176 p^{10} T^{26} + 7695 p^{11} T^{27} + 1100 p^{12} T^{28} + 441 p^{13} T^{29} + 51 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 8 T - 56 T^{2} - 1119 T^{3} - 2535 T^{4} + 53580 T^{5} + 414660 T^{6} - 444426 T^{7} - 545643 p T^{8} - 88385661 T^{9} + 648028965 T^{10} + 7245089643 T^{11} + 4155654153 T^{12} - 299678046864 T^{13} - 1552346203953 T^{14} + 5037849409691 T^{15} + 94963133561011 T^{16} + 5037849409691 p T^{17} - 1552346203953 p^{2} T^{18} - 299678046864 p^{3} T^{19} + 4155654153 p^{4} T^{20} + 7245089643 p^{5} T^{21} + 648028965 p^{6} T^{22} - 88385661 p^{7} T^{23} - 545643 p^{9} T^{24} - 444426 p^{9} T^{25} + 414660 p^{10} T^{26} + 53580 p^{11} T^{27} - 2535 p^{12} T^{28} - 1119 p^{13} T^{29} - 56 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 4 T - 79 T^{2} + 586 T^{3} + 3953 T^{4} - 34236 T^{5} - 106529 T^{6} + 829726 T^{7} + 7189374 T^{8} - 3863916 T^{9} - 380227312 T^{10} - 626737382 T^{11} + 19560478530 T^{12} - 6490482283 T^{13} - 475365247650 T^{14} - 345594392963 T^{15} + 17382311447355 T^{16} - 345594392963 p T^{17} - 475365247650 p^{2} T^{18} - 6490482283 p^{3} T^{19} + 19560478530 p^{4} T^{20} - 626737382 p^{5} T^{21} - 380227312 p^{6} T^{22} - 3863916 p^{7} T^{23} + 7189374 p^{8} T^{24} + 829726 p^{9} T^{25} - 106529 p^{10} T^{26} - 34236 p^{11} T^{27} + 3953 p^{12} T^{28} + 586 p^{13} T^{29} - 79 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 66 T + 2182 T^{2} + 48669 T^{3} + 831055 T^{4} + 11652366 T^{5} + 140245718 T^{6} + 1497553188 T^{7} + 14602228823 T^{8} + 133460258331 T^{9} + 1167497625859 T^{10} + 9901192379571 T^{11} + 81776547284693 T^{12} + 657352394939292 T^{13} + 5133967448395241 T^{14} + 38951463330774507 T^{15} + 287429954535186217 T^{16} + 38951463330774507 p T^{17} + 5133967448395241 p^{2} T^{18} + 657352394939292 p^{3} T^{19} + 81776547284693 p^{4} T^{20} + 9901192379571 p^{5} T^{21} + 1167497625859 p^{6} T^{22} + 133460258331 p^{7} T^{23} + 14602228823 p^{8} T^{24} + 1497553188 p^{9} T^{25} + 140245718 p^{10} T^{26} + 11652366 p^{11} T^{27} + 831055 p^{12} T^{28} + 48669 p^{13} T^{29} + 2182 p^{14} T^{30} + 66 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 51 T + 1392 T^{2} + 26652 T^{3} + 397514 T^{4} + 4868295 T^{5} + 50450316 T^{6} + 449005056 T^{7} + 3425533863 T^{8} + 21659931081 T^{9} + 98532123618 T^{10} + 53803714842 T^{11} - 5626751893175 T^{12} - 87843890832627 T^{13} - 955870402885209 T^{14} - 8747102777011068 T^{15} - 70916828475142069 T^{16} - 8747102777011068 p T^{17} - 955870402885209 p^{2} T^{18} - 87843890832627 p^{3} T^{19} - 5626751893175 p^{4} T^{20} + 53803714842 p^{5} T^{21} + 98532123618 p^{6} T^{22} + 21659931081 p^{7} T^{23} + 3425533863 p^{8} T^{24} + 449005056 p^{9} T^{25} + 50450316 p^{10} T^{26} + 4868295 p^{11} T^{27} + 397514 p^{12} T^{28} + 26652 p^{13} T^{29} + 1392 p^{14} T^{30} + 51 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 30 T + 776 T^{2} - 13665 T^{3} + 208005 T^{4} - 2593710 T^{5} + 28390399 T^{6} - 268109040 T^{7} + 2240276459 T^{8} - 268109040 p T^{9} + 28390399 p^{2} T^{10} - 2593710 p^{3} T^{11} + 208005 p^{4} T^{12} - 13665 p^{5} T^{13} + 776 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 13 T - 258 T^{2} + 4011 T^{3} + 35028 T^{4} - 644333 T^{5} - 51427 p T^{6} + 71925183 T^{7} + 280896507 T^{8} - 6225465362 T^{9} - 19715222826 T^{10} + 429031194656 T^{11} + 1171728043770 T^{12} - 21978520425855 T^{13} - 61484782388300 T^{14} + 555554083027185 T^{15} + 3540421883673161 T^{16} + 555554083027185 p T^{17} - 61484782388300 p^{2} T^{18} - 21978520425855 p^{3} T^{19} + 1171728043770 p^{4} T^{20} + 429031194656 p^{5} T^{21} - 19715222826 p^{6} T^{22} - 6225465362 p^{7} T^{23} + 280896507 p^{8} T^{24} + 71925183 p^{9} T^{25} - 51427 p^{11} T^{26} - 644333 p^{11} T^{27} + 35028 p^{12} T^{28} + 4011 p^{13} T^{29} - 258 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 24 T + 291 T^{2} + 1650 T^{3} - 3505 T^{4} - 170133 T^{5} - 1960992 T^{6} - 11101218 T^{7} - 10227345 T^{8} + 606977625 T^{9} + 6888210666 T^{10} + 59241878154 T^{11} + 428524037986 T^{12} + 2048662927665 T^{13} - 894626650320 T^{14} - 184623556051827 T^{15} - 1867719052729573 T^{16} - 184623556051827 p T^{17} - 894626650320 p^{2} T^{18} + 2048662927665 p^{3} T^{19} + 428524037986 p^{4} T^{20} + 59241878154 p^{5} T^{21} + 6888210666 p^{6} T^{22} + 606977625 p^{7} T^{23} - 10227345 p^{8} T^{24} - 11101218 p^{9} T^{25} - 1960992 p^{10} T^{26} - 170133 p^{11} T^{27} - 3505 p^{12} T^{28} + 1650 p^{13} T^{29} + 291 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 27 T + 139 T^{2} - 1876 T^{3} - 7260 T^{4} + 257335 T^{5} + 1029870 T^{6} - 17178229 T^{7} + 5250316 T^{8} + 1618213096 T^{9} - 4911646565 T^{10} - 112362177683 T^{11} + 689706510678 T^{12} + 3428876555724 T^{13} - 107385592794028 T^{14} - 256795178393646 T^{15} + 6542719574978891 T^{16} - 256795178393646 p T^{17} - 107385592794028 p^{2} T^{18} + 3428876555724 p^{3} T^{19} + 689706510678 p^{4} T^{20} - 112362177683 p^{5} T^{21} - 4911646565 p^{6} T^{22} + 1618213096 p^{7} T^{23} + 5250316 p^{8} T^{24} - 17178229 p^{9} T^{25} + 1029870 p^{10} T^{26} + 257335 p^{11} T^{27} - 7260 p^{12} T^{28} - 1876 p^{13} T^{29} + 139 p^{14} T^{30} + 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 8 T + 251 T^{2} + 1638 T^{3} + 27309 T^{4} + 101148 T^{5} + 656100 T^{6} - 10183886 T^{7} - 206180252 T^{8} - 2408277106 T^{9} - 28471539937 T^{10} - 2083375274 p T^{11} - 1002975234442 T^{12} + 4693136092136 T^{13} + 120901019964056 T^{14} + 1801972512171550 T^{15} + 17786739207624689 T^{16} + 1801972512171550 p T^{17} + 120901019964056 p^{2} T^{18} + 4693136092136 p^{3} T^{19} - 1002975234442 p^{4} T^{20} - 2083375274 p^{6} T^{21} - 28471539937 p^{6} T^{22} - 2408277106 p^{7} T^{23} - 206180252 p^{8} T^{24} - 10183886 p^{9} T^{25} + 656100 p^{10} T^{26} + 101148 p^{11} T^{27} + 27309 p^{12} T^{28} + 1638 p^{13} T^{29} + 251 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 4 T - 25 T^{2} + 1217 T^{3} + 13244 T^{4} + 5091 T^{5} + 1191835 T^{6} + 9775289 T^{7} - 48857298 T^{8} + 622367196 T^{9} + 7963035860 T^{10} - 98997330340 T^{11} - 530629557609 T^{12} + 2267011946350 T^{13} - 106966411977699 T^{14} - 866443992715393 T^{15} - 268637739467595 T^{16} - 866443992715393 p T^{17} - 106966411977699 p^{2} T^{18} + 2267011946350 p^{3} T^{19} - 530629557609 p^{4} T^{20} - 98997330340 p^{5} T^{21} + 7963035860 p^{6} T^{22} + 622367196 p^{7} T^{23} - 48857298 p^{8} T^{24} + 9775289 p^{9} T^{25} + 1191835 p^{10} T^{26} + 5091 p^{11} T^{27} + 13244 p^{12} T^{28} + 1217 p^{13} T^{29} - 25 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 26 T + 143 T^{2} - 1094 T^{3} + 6290 T^{4} + 492630 T^{5} + 4195573 T^{6} - 10521257 T^{7} - 141498288 T^{8} + 4126730694 T^{9} + 51492035501 T^{10} - 116586696947 T^{11} - 3818611811694 T^{12} + 17664638582087 T^{13} + 483095599499973 T^{14} - 358930347922835 T^{15} - 42727393274415357 T^{16} - 358930347922835 p T^{17} + 483095599499973 p^{2} T^{18} + 17664638582087 p^{3} T^{19} - 3818611811694 p^{4} T^{20} - 116586696947 p^{5} T^{21} + 51492035501 p^{6} T^{22} + 4126730694 p^{7} T^{23} - 141498288 p^{8} T^{24} - 10521257 p^{9} T^{25} + 4195573 p^{10} T^{26} + 492630 p^{11} T^{27} + 6290 p^{12} T^{28} - 1094 p^{13} T^{29} + 143 p^{14} T^{30} + 26 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 37 T + 353 T^{2} - 4107 T^{3} - 109371 T^{4} - 370836 T^{5} + 11851737 T^{6} + 134206862 T^{7} - 442204397 T^{8} - 17577172970 T^{9} - 63138303652 T^{10} + 1337862004970 T^{11} + 12947461844639 T^{12} - 46465284959771 T^{13} - 1169620344005929 T^{14} - 216403698441325 T^{15} + 84467647825567967 T^{16} - 216403698441325 p T^{17} - 1169620344005929 p^{2} T^{18} - 46465284959771 p^{3} T^{19} + 12947461844639 p^{4} T^{20} + 1337862004970 p^{5} T^{21} - 63138303652 p^{6} T^{22} - 17577172970 p^{7} T^{23} - 442204397 p^{8} T^{24} + 134206862 p^{9} T^{25} + 11851737 p^{10} T^{26} - 370836 p^{11} T^{27} - 109371 p^{12} T^{28} - 4107 p^{13} T^{29} + 353 p^{14} T^{30} + 37 p^{15} T^{31} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.62270399994028239883646825947, −2.50370061639270252552493513804, −2.50069198562671208188312319901, −2.48152815117263168133253794093, −2.33586002978878400633317908186, −2.25681887305591104620293713093, −2.23033955034778538435665881132, −2.16324586879743578749384746307, −1.85156383072468032054021921576, −1.76137000115694517783896238306, −1.59862063129748964885019803293, −1.51478448855232530386434619079, −1.49247343118003240854163686680, −1.42781109226588996945734676493, −1.42518020875102605951628098711, −1.40001502087530718483431584353, −1.28893563432326114625570907025, −1.28321254598936101035369515758, −1.25118120754862596623467659603, −1.19760498993459381065031113114, −1.09939368500931177266664979634, −0.55575984970366626800526093875, −0.50942365787679508989838127613, −0.19021757285398163163520083963, −0.00539520466278984601656687743, 0.00539520466278984601656687743, 0.19021757285398163163520083963, 0.50942365787679508989838127613, 0.55575984970366626800526093875, 1.09939368500931177266664979634, 1.19760498993459381065031113114, 1.25118120754862596623467659603, 1.28321254598936101035369515758, 1.28893563432326114625570907025, 1.40001502087530718483431584353, 1.42518020875102605951628098711, 1.42781109226588996945734676493, 1.49247343118003240854163686680, 1.51478448855232530386434619079, 1.59862063129748964885019803293, 1.76137000115694517783896238306, 1.85156383072468032054021921576, 2.16324586879743578749384746307, 2.23033955034778538435665881132, 2.25681887305591104620293713093, 2.33586002978878400633317908186, 2.48152815117263168133253794093, 2.50069198562671208188312319901, 2.50370061639270252552493513804, 2.62270399994028239883646825947
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.