Dirichlet series
| L(s) = 1 | − 4·2-s − 2·3-s + 4-s − 4·5-s + 8·6-s − 2·7-s + 16·8-s − 17·9-s + 16·10-s − 14·11-s − 2·12-s + 8·14-s + 8·15-s − 21·16-s + 4·17-s + 68·18-s − 22·19-s − 4·20-s + 4·21-s + 56·22-s + 4·23-s − 32·24-s − 33·25-s + 42·27-s − 2·28-s − 8·29-s − 32·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 0.755·7-s + 5.65·8-s − 5.66·9-s + 5.05·10-s − 4.22·11-s − 0.577·12-s + 2.13·14-s + 2.06·15-s − 5.25·16-s + 0.970·17-s + 16.0·18-s − 5.04·19-s − 0.894·20-s + 0.872·21-s + 11.9·22-s + 0.834·23-s − 6.53·24-s − 6.59·25-s + 8.08·27-s − 0.377·28-s − 1.48·29-s − 5.84·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(13^{32} \cdot 31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.79848\times 10^{25}\) |
| Root analytic conductor: | \(6.46789\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 13^{32} \cdot 31^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
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 | 13 | \( 1 \) |
| 31 | \( ( 1 - T )^{16} \) | |
| good | 2 | \( 1 + p^{2} T + 15 T^{2} + 5 p^{3} T^{3} + 51 p T^{4} + 57 p^{2} T^{5} + 243 p T^{6} + 479 p T^{7} + 909 p T^{8} + 1633 p T^{9} + 2851 p T^{10} + 2375 p^{2} T^{11} + 7723 p T^{12} + 12055 p T^{13} + 18399 p T^{14} + 6767 p^{3} T^{15} + 77909 T^{16} + 6767 p^{4} T^{17} + 18399 p^{3} T^{18} + 12055 p^{4} T^{19} + 7723 p^{5} T^{20} + 2375 p^{7} T^{21} + 2851 p^{7} T^{22} + 1633 p^{8} T^{23} + 909 p^{9} T^{24} + 479 p^{10} T^{25} + 243 p^{11} T^{26} + 57 p^{13} T^{27} + 51 p^{13} T^{28} + 5 p^{16} T^{29} + 15 p^{14} T^{30} + p^{17} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + 2 T + 7 p T^{2} + 34 T^{3} + 218 T^{4} + 100 p T^{5} + 1555 T^{6} + 628 p T^{7} + 8731 T^{8} + 9520 T^{9} + 41144 T^{10} + 41066 T^{11} + 167989 T^{12} + 155486 T^{13} + 603142 T^{14} + 522428 T^{15} + 1919026 T^{16} + 522428 p T^{17} + 603142 p^{2} T^{18} + 155486 p^{3} T^{19} + 167989 p^{4} T^{20} + 41066 p^{5} T^{21} + 41144 p^{6} T^{22} + 9520 p^{7} T^{23} + 8731 p^{8} T^{24} + 628 p^{10} T^{25} + 1555 p^{10} T^{26} + 100 p^{12} T^{27} + 218 p^{12} T^{28} + 34 p^{13} T^{29} + 7 p^{15} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 + 4 T + 49 T^{2} + 34 p T^{3} + 1169 T^{4} + 708 p T^{5} + 18017 T^{6} + 48152 T^{7} + 201957 T^{8} + 482166 T^{9} + 70622 p^{2} T^{10} + 3812784 T^{11} + 12648819 T^{12} + 25046692 T^{13} + 77121452 T^{14} + 141984952 T^{15} + 410664657 T^{16} + 141984952 p T^{17} + 77121452 p^{2} T^{18} + 25046692 p^{3} T^{19} + 12648819 p^{4} T^{20} + 3812784 p^{5} T^{21} + 70622 p^{8} T^{22} + 482166 p^{7} T^{23} + 201957 p^{8} T^{24} + 48152 p^{9} T^{25} + 18017 p^{10} T^{26} + 708 p^{12} T^{27} + 1169 p^{12} T^{28} + 34 p^{14} T^{29} + 49 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 2 T + 6 p T^{2} + 10 p T^{3} + 850 T^{4} + 1110 T^{5} + 10960 T^{6} + 9916 T^{7} + 101646 T^{8} + 41362 T^{9} + 719683 T^{10} - 249098 T^{11} + 3929080 T^{12} - 6823952 T^{13} + 16827506 T^{14} - 74650080 T^{15} + 82389777 T^{16} - 74650080 p T^{17} + 16827506 p^{2} T^{18} - 6823952 p^{3} T^{19} + 3929080 p^{4} T^{20} - 249098 p^{5} T^{21} + 719683 p^{6} T^{22} + 41362 p^{7} T^{23} + 101646 p^{8} T^{24} + 9916 p^{9} T^{25} + 10960 p^{10} T^{26} + 1110 p^{11} T^{27} + 850 p^{12} T^{28} + 10 p^{14} T^{29} + 6 p^{15} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 14 T + 193 T^{2} + 1766 T^{3} + 15042 T^{4} + 106000 T^{5} + 696396 T^{6} + 369790 p T^{7} + 22319681 T^{8} + 112387838 T^{9} + 535177240 T^{10} + 2376412498 T^{11} + 10027317573 T^{12} + 39793437236 T^{13} + 150520146935 T^{14} + 537782186286 T^{15} + 1834017527270 T^{16} + 537782186286 p T^{17} + 150520146935 p^{2} T^{18} + 39793437236 p^{3} T^{19} + 10027317573 p^{4} T^{20} + 2376412498 p^{5} T^{21} + 535177240 p^{6} T^{22} + 112387838 p^{7} T^{23} + 22319681 p^{8} T^{24} + 369790 p^{10} T^{25} + 696396 p^{10} T^{26} + 106000 p^{11} T^{27} + 15042 p^{12} T^{28} + 1766 p^{13} T^{29} + 193 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 4 T + 167 T^{2} - 638 T^{3} + 14230 T^{4} - 51436 T^{5} + 811354 T^{6} - 2759804 T^{7} + 34419993 T^{8} - 109711108 T^{9} + 1147272314 T^{10} - 3412018000 T^{11} + 1823445881 p T^{12} - 85579508576 T^{13} + 691428574449 T^{14} - 1760474326102 T^{15} + 12855128851858 T^{16} - 1760474326102 p T^{17} + 691428574449 p^{2} T^{18} - 85579508576 p^{3} T^{19} + 1823445881 p^{5} T^{20} - 3412018000 p^{5} T^{21} + 1147272314 p^{6} T^{22} - 109711108 p^{7} T^{23} + 34419993 p^{8} T^{24} - 2759804 p^{9} T^{25} + 811354 p^{10} T^{26} - 51436 p^{11} T^{27} + 14230 p^{12} T^{28} - 638 p^{13} T^{29} + 167 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 22 T + 423 T^{2} + 290 p T^{3} + 3396 p T^{4} + 622948 T^{5} + 5538440 T^{6} + 43415552 T^{7} + 318418086 T^{8} + 2126188226 T^{9} + 13422249200 T^{10} + 78519921788 T^{11} + 437210858146 T^{12} + 2278943534750 T^{13} + 11352944126987 T^{14} + 53226592711684 T^{15} + 238957361874987 T^{16} + 53226592711684 p T^{17} + 11352944126987 p^{2} T^{18} + 2278943534750 p^{3} T^{19} + 437210858146 p^{4} T^{20} + 78519921788 p^{5} T^{21} + 13422249200 p^{6} T^{22} + 2126188226 p^{7} T^{23} + 318418086 p^{8} T^{24} + 43415552 p^{9} T^{25} + 5538440 p^{10} T^{26} + 622948 p^{11} T^{27} + 3396 p^{13} T^{28} + 290 p^{14} T^{29} + 423 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 4 T + 182 T^{2} - 876 T^{3} + 734 p T^{4} - 85596 T^{5} + 1066996 T^{6} - 5179642 T^{7} + 50683499 T^{8} - 225557226 T^{9} + 1897105692 T^{10} - 7709902006 T^{11} + 58547822885 T^{12} - 219399866216 T^{13} + 1565541834150 T^{14} - 5489823491962 T^{15} + 37688475152030 T^{16} - 5489823491962 p T^{17} + 1565541834150 p^{2} T^{18} - 219399866216 p^{3} T^{19} + 58547822885 p^{4} T^{20} - 7709902006 p^{5} T^{21} + 1897105692 p^{6} T^{22} - 225557226 p^{7} T^{23} + 50683499 p^{8} T^{24} - 5179642 p^{9} T^{25} + 1066996 p^{10} T^{26} - 85596 p^{11} T^{27} + 734 p^{13} T^{28} - 876 p^{13} T^{29} + 182 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 8 T + 220 T^{2} + 1110 T^{3} + 21310 T^{4} + 68526 T^{5} + 1342857 T^{6} + 2179216 T^{7} + 63203332 T^{8} - 1461056 T^{9} + 2417825971 T^{10} - 4095947054 T^{11} + 80860738337 T^{12} - 256714762740 T^{13} + 2493206148736 T^{14} - 10114188632182 T^{15} + 73571341464604 T^{16} - 10114188632182 p T^{17} + 2493206148736 p^{2} T^{18} - 256714762740 p^{3} T^{19} + 80860738337 p^{4} T^{20} - 4095947054 p^{5} T^{21} + 2417825971 p^{6} T^{22} - 1461056 p^{7} T^{23} + 63203332 p^{8} T^{24} + 2179216 p^{9} T^{25} + 1342857 p^{10} T^{26} + 68526 p^{11} T^{27} + 21310 p^{12} T^{28} + 1110 p^{13} T^{29} + 220 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 16 T + 281 T^{2} + 3018 T^{3} + 34927 T^{4} + 313190 T^{5} + 2884151 T^{6} + 22212378 T^{7} + 173346155 T^{8} + 1184293678 T^{9} + 8159404947 T^{10} + 50203096966 T^{11} + 314774277836 T^{12} + 1809673031534 T^{13} + 10920687079539 T^{14} + 62095847832320 T^{15} + 384596080229138 T^{16} + 62095847832320 p T^{17} + 10920687079539 p^{2} T^{18} + 1809673031534 p^{3} T^{19} + 314774277836 p^{4} T^{20} + 50203096966 p^{5} T^{21} + 8159404947 p^{6} T^{22} + 1184293678 p^{7} T^{23} + 173346155 p^{8} T^{24} + 22212378 p^{9} T^{25} + 2884151 p^{10} T^{26} + 313190 p^{11} T^{27} + 34927 p^{12} T^{28} + 3018 p^{13} T^{29} + 281 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 44 T + 1267 T^{2} + 27010 T^{3} + 474658 T^{4} + 7134028 T^{5} + 94746078 T^{6} + 1131504052 T^{7} + 12335577335 T^{8} + 123970933278 T^{9} + 1158128963871 T^{10} + 10116089856576 T^{11} + 83030611047783 T^{12} + 642594586767924 T^{13} + 4702649197331611 T^{14} + 32598947411850872 T^{15} + 5227179884423777 p T^{16} + 32598947411850872 p T^{17} + 4702649197331611 p^{2} T^{18} + 642594586767924 p^{3} T^{19} + 83030611047783 p^{4} T^{20} + 10116089856576 p^{5} T^{21} + 1158128963871 p^{6} T^{22} + 123970933278 p^{7} T^{23} + 12335577335 p^{8} T^{24} + 1131504052 p^{9} T^{25} + 94746078 p^{10} T^{26} + 7134028 p^{11} T^{27} + 474658 p^{12} T^{28} + 27010 p^{13} T^{29} + 1267 p^{14} T^{30} + 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 16 T + 491 T^{2} - 6320 T^{3} + 2631 p T^{4} - 1240932 T^{5} + 16691615 T^{6} - 161301798 T^{7} + 1792302122 T^{8} - 15573445800 T^{9} + 149872010422 T^{10} - 1185239906190 T^{11} + 10146060020032 T^{12} - 73496852836876 T^{13} + 568551289690556 T^{14} - 3780355123235520 T^{15} + 26672834634296236 T^{16} - 3780355123235520 p T^{17} + 568551289690556 p^{2} T^{18} - 73496852836876 p^{3} T^{19} + 10146060020032 p^{4} T^{20} - 1185239906190 p^{5} T^{21} + 149872010422 p^{6} T^{22} - 15573445800 p^{7} T^{23} + 1792302122 p^{8} T^{24} - 161301798 p^{9} T^{25} + 16691615 p^{10} T^{26} - 1240932 p^{11} T^{27} + 2631 p^{13} T^{28} - 6320 p^{13} T^{29} + 491 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 10 T + 319 T^{2} + 2158 T^{3} + 45197 T^{4} + 227486 T^{5} + 4372173 T^{6} + 17808894 T^{7} + 345920114 T^{8} + 1191969258 T^{9} + 23616948612 T^{10} + 70926148726 T^{11} + 1429263970430 T^{12} + 3855128819978 T^{13} + 77579226337384 T^{14} + 192114798287706 T^{15} + 3812710321024804 T^{16} + 192114798287706 p T^{17} + 77579226337384 p^{2} T^{18} + 3855128819978 p^{3} T^{19} + 1429263970430 p^{4} T^{20} + 70926148726 p^{5} T^{21} + 23616948612 p^{6} T^{22} + 1191969258 p^{7} T^{23} + 345920114 p^{8} T^{24} + 17808894 p^{9} T^{25} + 4372173 p^{10} T^{26} + 227486 p^{11} T^{27} + 45197 p^{12} T^{28} + 2158 p^{13} T^{29} + 319 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 6 T + 469 T^{2} + 3382 T^{3} + 113080 T^{4} + 910192 T^{5} + 18630434 T^{6} + 157418214 T^{7} + 2337521583 T^{8} + 19768857434 T^{9} + 234972166028 T^{10} + 1922908429030 T^{11} + 19415354023723 T^{12} + 150287264202844 T^{13} + 1336877045345107 T^{14} + 9629706833693278 T^{15} + 77256050069348266 T^{16} + 9629706833693278 p T^{17} + 1336877045345107 p^{2} T^{18} + 150287264202844 p^{3} T^{19} + 19415354023723 p^{4} T^{20} + 1922908429030 p^{5} T^{21} + 234972166028 p^{6} T^{22} + 19768857434 p^{7} T^{23} + 2337521583 p^{8} T^{24} + 157418214 p^{9} T^{25} + 18630434 p^{10} T^{26} + 910192 p^{11} T^{27} + 113080 p^{12} T^{28} + 3382 p^{13} T^{29} + 469 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 2 T + 311 T^{2} + 280 T^{3} + 56871 T^{4} + 27892 T^{5} + 7683267 T^{6} + 3616286 T^{7} + 836116241 T^{8} + 479405556 T^{9} + 76215312886 T^{10} + 55605533344 T^{11} + 6016831986079 T^{12} + 5203987075418 T^{13} + 419411701854944 T^{14} + 386178217640108 T^{15} + 26118253198678209 T^{16} + 386178217640108 p T^{17} + 419411701854944 p^{2} T^{18} + 5203987075418 p^{3} T^{19} + 6016831986079 p^{4} T^{20} + 55605533344 p^{5} T^{21} + 76215312886 p^{6} T^{22} + 479405556 p^{7} T^{23} + 836116241 p^{8} T^{24} + 3616286 p^{9} T^{25} + 7683267 p^{10} T^{26} + 27892 p^{11} T^{27} + 56871 p^{12} T^{28} + 280 p^{13} T^{29} + 311 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 8 T + 605 T^{2} - 4160 T^{3} + 179090 T^{4} - 1077090 T^{5} + 34658063 T^{6} - 184200124 T^{7} + 4937753413 T^{8} - 23384070246 T^{9} + 552964363604 T^{10} - 2353709635954 T^{11} + 50716482075547 T^{12} - 195982769600702 T^{13} + 3911640281431720 T^{14} - 13871953720143360 T^{15} + 257587502471292710 T^{16} - 13871953720143360 p T^{17} + 3911640281431720 p^{2} T^{18} - 195982769600702 p^{3} T^{19} + 50716482075547 p^{4} T^{20} - 2353709635954 p^{5} T^{21} + 552964363604 p^{6} T^{22} - 23384070246 p^{7} T^{23} + 4937753413 p^{8} T^{24} - 184200124 p^{9} T^{25} + 34658063 p^{10} T^{26} - 1077090 p^{11} T^{27} + 179090 p^{12} T^{28} - 4160 p^{13} T^{29} + 605 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 8 T + 540 T^{2} - 4000 T^{3} + 150116 T^{4} - 1042956 T^{5} + 28529992 T^{6} - 187196214 T^{7} + 4144141138 T^{8} - 25724621186 T^{9} + 487341620837 T^{10} - 2857205859716 T^{11} + 47936582041667 T^{12} - 264447903444180 T^{13} + 4018219941545291 T^{14} - 20744677924238584 T^{15} + 289809615272188020 T^{16} - 20744677924238584 p T^{17} + 4018219941545291 p^{2} T^{18} - 264447903444180 p^{3} T^{19} + 47936582041667 p^{4} T^{20} - 2857205859716 p^{5} T^{21} + 487341620837 p^{6} T^{22} - 25724621186 p^{7} T^{23} + 4144141138 p^{8} T^{24} - 187196214 p^{9} T^{25} + 28529992 p^{10} T^{26} - 1042956 p^{11} T^{27} + 150116 p^{12} T^{28} - 4000 p^{13} T^{29} + 540 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 50 T + 2023 T^{2} + 57796 T^{3} + 1435378 T^{4} + 29942746 T^{5} + 562520724 T^{6} + 9392394082 T^{7} + 144049911404 T^{8} + 2015332844396 T^{9} + 26207157566060 T^{10} + 315243738983424 T^{11} + 3550204210177108 T^{12} + 37273943292315142 T^{13} + 367928198923001699 T^{14} + 3399544170131210334 T^{15} + 29588800057516388083 T^{16} + 3399544170131210334 p T^{17} + 367928198923001699 p^{2} T^{18} + 37273943292315142 p^{3} T^{19} + 3550204210177108 p^{4} T^{20} + 315243738983424 p^{5} T^{21} + 26207157566060 p^{6} T^{22} + 2015332844396 p^{7} T^{23} + 144049911404 p^{8} T^{24} + 9392394082 p^{9} T^{25} + 562520724 p^{10} T^{26} + 29942746 p^{11} T^{27} + 1435378 p^{12} T^{28} + 57796 p^{13} T^{29} + 2023 p^{14} T^{30} + 50 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 14 T + 728 T^{2} + 10292 T^{3} + 272102 T^{4} + 3665304 T^{5} + 68341783 T^{6} + 849758052 T^{7} + 12736535884 T^{8} + 144527743928 T^{9} + 1853079469817 T^{10} + 19174355415656 T^{11} + 217117284321741 T^{12} + 2053154612783592 T^{13} + 20894461478312908 T^{14} + 180776600638000586 T^{15} + 1670393972000490008 T^{16} + 180776600638000586 p T^{17} + 20894461478312908 p^{2} T^{18} + 2053154612783592 p^{3} T^{19} + 217117284321741 p^{4} T^{20} + 19174355415656 p^{5} T^{21} + 1853079469817 p^{6} T^{22} + 144527743928 p^{7} T^{23} + 12736535884 p^{8} T^{24} + 849758052 p^{9} T^{25} + 68341783 p^{10} T^{26} + 3665304 p^{11} T^{27} + 272102 p^{12} T^{28} + 10292 p^{13} T^{29} + 728 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 32 T + 1315 T^{2} - 30182 T^{3} + 9297 p T^{4} - 13343174 T^{5} + 243579549 T^{6} - 3692680710 T^{7} + 55185674243 T^{8} - 721646400906 T^{9} + 9239431689267 T^{10} - 106654598620084 T^{11} + 1203548318055452 T^{12} - 12472310267258742 T^{13} + 126429719218540843 T^{14} - 1189380926010068494 T^{15} + 10954443490430363522 T^{16} - 1189380926010068494 p T^{17} + 126429719218540843 p^{2} T^{18} - 12472310267258742 p^{3} T^{19} + 1203548318055452 p^{4} T^{20} - 106654598620084 p^{5} T^{21} + 9239431689267 p^{6} T^{22} - 721646400906 p^{7} T^{23} + 55185674243 p^{8} T^{24} - 3692680710 p^{9} T^{25} + 243579549 p^{10} T^{26} - 13343174 p^{11} T^{27} + 9297 p^{13} T^{28} - 30182 p^{13} T^{29} + 1315 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 20 T + 817 T^{2} - 12646 T^{3} + 315387 T^{4} - 4128658 T^{5} + 80006467 T^{6} - 924494530 T^{7} + 15177213168 T^{8} - 158382621132 T^{9} + 2295866758492 T^{10} - 21922133767156 T^{11} + 286991147048010 T^{12} - 2525615695552308 T^{13} + 30248822567620700 T^{14} - 246118235375152034 T^{15} + 2715969577494055212 T^{16} - 246118235375152034 p T^{17} + 30248822567620700 p^{2} T^{18} - 2525615695552308 p^{3} T^{19} + 286991147048010 p^{4} T^{20} - 21922133767156 p^{5} T^{21} + 2295866758492 p^{6} T^{22} - 158382621132 p^{7} T^{23} + 15177213168 p^{8} T^{24} - 924494530 p^{9} T^{25} + 80006467 p^{10} T^{26} - 4128658 p^{11} T^{27} + 315387 p^{12} T^{28} - 12646 p^{13} T^{29} + 817 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 52 T + 2233 T^{2} + 65480 T^{3} + 1673018 T^{4} + 34959414 T^{5} + 655417961 T^{6} + 10652368896 T^{7} + 158010711225 T^{8} + 2086163588344 T^{9} + 25456627659406 T^{10} + 281146798817394 T^{11} + 2919262491032063 T^{12} + 28071678826415070 T^{13} + 263476173129907260 T^{14} + 2398723875918036890 T^{15} + 22562710182142469814 T^{16} + 2398723875918036890 p T^{17} + 263476173129907260 p^{2} T^{18} + 28071678826415070 p^{3} T^{19} + 2919262491032063 p^{4} T^{20} + 281146798817394 p^{5} T^{21} + 25456627659406 p^{6} T^{22} + 2086163588344 p^{7} T^{23} + 158010711225 p^{8} T^{24} + 10652368896 p^{9} T^{25} + 655417961 p^{10} T^{26} + 34959414 p^{11} T^{27} + 1673018 p^{12} T^{28} + 65480 p^{13} T^{29} + 2233 p^{14} T^{30} + 52 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 18 T + 867 T^{2} + 14750 T^{3} + 390563 T^{4} + 6189082 T^{5} + 119716322 T^{6} + 1751108066 T^{7} + 27705709782 T^{8} + 372138728000 T^{9} + 5100351015684 T^{10} + 62793193346864 T^{11} + 769145311830846 T^{12} + 8682674022988166 T^{13} + 96645487447424545 T^{14} + 1001500160242145840 T^{15} + 10209699886837918733 T^{16} + 1001500160242145840 p T^{17} + 96645487447424545 p^{2} T^{18} + 8682674022988166 p^{3} T^{19} + 769145311830846 p^{4} T^{20} + 62793193346864 p^{5} T^{21} + 5100351015684 p^{6} T^{22} + 372138728000 p^{7} T^{23} + 27705709782 p^{8} T^{24} + 1751108066 p^{9} T^{25} + 119716322 p^{10} T^{26} + 6189082 p^{11} T^{27} + 390563 p^{12} T^{28} + 14750 p^{13} T^{29} + 867 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.32143578457943798531059622217, −2.30494045029715226801806104331, −2.30288182914904230313940328105, −2.24681407803611248892888574596, −2.24532660892996888005665762445, −2.15327530666983735484896126536, −2.12374129598787968483439628076, −2.05952366243768237780797924130, −1.97345510662610369746520035152, −1.82070111567902837298306923765, −1.72316156480862022312872760906, −1.71282293219670442217283631792, −1.69148373532210599816435579608, −1.59529824775446997348684160442, −1.45480559252618530289291371430, −1.39646247479877061944375488863, −1.26593893233984783363206433157, −1.23662272798479993923251349145, −1.18408596888953319598875137964, −1.16009418575403076540459672906, −1.00810781733585479263643845183, −0.984414629076182756252752061263, −0.802386340927262768875052388622, −0.74843859833453218183935693036, −0.72007876663661345669855939593, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.72007876663661345669855939593, 0.74843859833453218183935693036, 0.802386340927262768875052388622, 0.984414629076182756252752061263, 1.00810781733585479263643845183, 1.16009418575403076540459672906, 1.18408596888953319598875137964, 1.23662272798479993923251349145, 1.26593893233984783363206433157, 1.39646247479877061944375488863, 1.45480559252618530289291371430, 1.59529824775446997348684160442, 1.69148373532210599816435579608, 1.71282293219670442217283631792, 1.72316156480862022312872760906, 1.82070111567902837298306923765, 1.97345510662610369746520035152, 2.05952366243768237780797924130, 2.12374129598787968483439628076, 2.15327530666983735484896126536, 2.24532660892996888005665762445, 2.24681407803611248892888574596, 2.30288182914904230313940328105, 2.30494045029715226801806104331, 2.32143578457943798531059622217
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.