Dirichlet series
| L(s) = 1 | − 2·2-s − 16·3-s − 4-s − 16·5-s + 32·6-s − 2·7-s + 3·8-s + 136·9-s + 32·10-s + 3·11-s + 16·12-s − 16·13-s + 4·14-s + 256·15-s − 16-s − 13·17-s − 272·18-s + 16·20-s + 32·21-s − 6·22-s − 15·23-s − 48·24-s + 136·25-s + 32·26-s − 816·27-s + 2·28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 9.23·3-s − 1/2·4-s − 7.15·5-s + 13.0·6-s − 0.755·7-s + 1.06·8-s + 45.3·9-s + 10.1·10-s + 0.904·11-s + 4.61·12-s − 4.43·13-s + 1.06·14-s + 66.0·15-s − 1/4·16-s − 3.15·17-s − 64.1·18-s + 3.57·20-s + 6.98·21-s − 1.27·22-s − 3.12·23-s − 9.79·24-s + 27.1·25-s + 6.27·26-s − 157.·27-s + 0.377·28-s − 0.742·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 13^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 13^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{16} \cdot 13^{16} \cdot 31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.68509\times 10^{26}\) |
| Root analytic conductor: | \(6.94763\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{16} \cdot 13^{16} \cdot 31^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0002208269411\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0002208269411\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{16} \) |
| 5 | \( ( 1 + T )^{16} \) | |
| 13 | \( ( 1 + T )^{16} \) | |
| 31 | \( ( 1 - T )^{16} \) | |
| good | 2 | \( 1 + p T + 5 T^{2} + 9 T^{3} + 9 p T^{4} + 31 T^{5} + 51 T^{6} + p^{6} T^{7} + 3 p^{5} T^{8} + 111 T^{9} + 75 p T^{10} + 143 T^{11} + 163 T^{12} + 77 p T^{13} + 65 p^{2} T^{14} + 75 p T^{15} + 91 p^{2} T^{16} + 75 p^{2} T^{17} + 65 p^{4} T^{18} + 77 p^{4} T^{19} + 163 p^{4} T^{20} + 143 p^{5} T^{21} + 75 p^{7} T^{22} + 111 p^{7} T^{23} + 3 p^{13} T^{24} + p^{15} T^{25} + 51 p^{10} T^{26} + 31 p^{11} T^{27} + 9 p^{13} T^{28} + 9 p^{13} T^{29} + 5 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 + 2 T + 50 T^{2} + 89 T^{3} + 1210 T^{4} + 2048 T^{5} + 19143 T^{6} + 31768 T^{7} + 225824 T^{8} + 366466 T^{9} + 2174583 T^{10} + 3335825 T^{11} + 18362576 T^{12} + 3640528 p T^{13} + 142396848 T^{14} + 178561458 T^{15} + 1030951434 T^{16} + 178561458 p T^{17} + 142396848 p^{2} T^{18} + 3640528 p^{4} T^{19} + 18362576 p^{4} T^{20} + 3335825 p^{5} T^{21} + 2174583 p^{6} T^{22} + 366466 p^{7} T^{23} + 225824 p^{8} T^{24} + 31768 p^{9} T^{25} + 19143 p^{10} T^{26} + 2048 p^{11} T^{27} + 1210 p^{12} T^{28} + 89 p^{13} T^{29} + 50 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 3 T + 64 T^{2} - 193 T^{3} + 2243 T^{4} - 6366 T^{5} + 55970 T^{6} - 147442 T^{7} + 1102389 T^{8} - 2675055 T^{9} + 18139937 T^{10} - 40407234 T^{11} + 23479911 p T^{12} - 533591103 T^{13} + 3274491765 T^{14} - 6369595726 T^{15} + 37694876884 T^{16} - 6369595726 p T^{17} + 3274491765 p^{2} T^{18} - 533591103 p^{3} T^{19} + 23479911 p^{5} T^{20} - 40407234 p^{5} T^{21} + 18139937 p^{6} T^{22} - 2675055 p^{7} T^{23} + 1102389 p^{8} T^{24} - 147442 p^{9} T^{25} + 55970 p^{10} T^{26} - 6366 p^{11} T^{27} + 2243 p^{12} T^{28} - 193 p^{13} T^{29} + 64 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 13 T + 193 T^{2} + 1712 T^{3} + 15711 T^{4} + 109513 T^{5} + 777507 T^{6} + 269320 p T^{7} + 27577189 T^{8} + 144105714 T^{9} + 777444113 T^{10} + 3732847661 T^{11} + 18605727173 T^{12} + 288927786 p^{2} T^{13} + 388511545035 T^{14} + 95994920633 p T^{15} + 7081862672620 T^{16} + 95994920633 p^{2} T^{17} + 388511545035 p^{2} T^{18} + 288927786 p^{5} T^{19} + 18605727173 p^{4} T^{20} + 3732847661 p^{5} T^{21} + 777444113 p^{6} T^{22} + 144105714 p^{7} T^{23} + 27577189 p^{8} T^{24} + 269320 p^{10} T^{25} + 777507 p^{10} T^{26} + 109513 p^{11} T^{27} + 15711 p^{12} T^{28} + 1712 p^{13} T^{29} + 193 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 109 T^{2} + 106 T^{3} + 6675 T^{4} + 8414 T^{5} + 299350 T^{6} + 379536 T^{7} + 10587609 T^{8} + 12343192 T^{9} + 310790901 T^{10} + 316459728 T^{11} + 7847637933 T^{12} + 6841442614 T^{13} + 174911214076 T^{14} + 135205903146 T^{15} + 3495894602372 T^{16} + 135205903146 p T^{17} + 174911214076 p^{2} T^{18} + 6841442614 p^{3} T^{19} + 7847637933 p^{4} T^{20} + 316459728 p^{5} T^{21} + 310790901 p^{6} T^{22} + 12343192 p^{7} T^{23} + 10587609 p^{8} T^{24} + 379536 p^{9} T^{25} + 299350 p^{10} T^{26} + 8414 p^{11} T^{27} + 6675 p^{12} T^{28} + 106 p^{13} T^{29} + 109 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 15 T + 306 T^{2} + 3566 T^{3} + 43798 T^{4} + 420396 T^{5} + 3964002 T^{6} + 32491786 T^{7} + 256094042 T^{8} + 1835588331 T^{9} + 546978415 p T^{10} + 80100713367 T^{11} + 487384522608 T^{12} + 2784477648893 T^{13} + 15212518533859 T^{14} + 78432479998012 T^{15} + 386912926546366 T^{16} + 78432479998012 p T^{17} + 15212518533859 p^{2} T^{18} + 2784477648893 p^{3} T^{19} + 487384522608 p^{4} T^{20} + 80100713367 p^{5} T^{21} + 546978415 p^{7} T^{22} + 1835588331 p^{7} T^{23} + 256094042 p^{8} T^{24} + 32491786 p^{9} T^{25} + 3964002 p^{10} T^{26} + 420396 p^{11} T^{27} + 43798 p^{12} T^{28} + 3566 p^{13} T^{29} + 306 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 4 T + 224 T^{2} + 1032 T^{3} + 26377 T^{4} + 129339 T^{5} + 2166768 T^{6} + 10741132 T^{7} + 137318274 T^{8} + 668819972 T^{9} + 7057042136 T^{10} + 33173061626 T^{11} + 302876690061 T^{12} + 1355231071284 T^{13} + 11043413752920 T^{14} + 1602033447195 p T^{15} + 345296526163294 T^{16} + 1602033447195 p^{2} T^{17} + 11043413752920 p^{2} T^{18} + 1355231071284 p^{3} T^{19} + 302876690061 p^{4} T^{20} + 33173061626 p^{5} T^{21} + 7057042136 p^{6} T^{22} + 668819972 p^{7} T^{23} + 137318274 p^{8} T^{24} + 10741132 p^{9} T^{25} + 2166768 p^{10} T^{26} + 129339 p^{11} T^{27} + 26377 p^{12} T^{28} + 1032 p^{13} T^{29} + 224 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 12 T + 324 T^{2} - 2600 T^{3} + 42619 T^{4} - 236790 T^{5} + 3331569 T^{6} - 12784312 T^{7} + 199733828 T^{8} - 554058649 T^{9} + 10901926185 T^{10} - 23755864697 T^{11} + 537408035653 T^{12} - 879086288177 T^{13} + 22725853891322 T^{14} - 26774036511451 T^{15} + 863896234214934 T^{16} - 26774036511451 p T^{17} + 22725853891322 p^{2} T^{18} - 879086288177 p^{3} T^{19} + 537408035653 p^{4} T^{20} - 23755864697 p^{5} T^{21} + 10901926185 p^{6} T^{22} - 554058649 p^{7} T^{23} + 199733828 p^{8} T^{24} - 12784312 p^{9} T^{25} + 3331569 p^{10} T^{26} - 236790 p^{11} T^{27} + 42619 p^{12} T^{28} - 2600 p^{13} T^{29} + 324 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 12 T + 213 T^{2} + 2226 T^{3} + 27055 T^{4} + 261025 T^{5} + 2682534 T^{6} + 23295398 T^{7} + 210198493 T^{8} + 1689153318 T^{9} + 13921663582 T^{10} + 104462653352 T^{11} + 794098394186 T^{12} + 5556589216666 T^{13} + 39399979216673 T^{14} + 259567431920235 T^{15} + 1723993243436030 T^{16} + 259567431920235 p T^{17} + 39399979216673 p^{2} T^{18} + 5556589216666 p^{3} T^{19} + 794098394186 p^{4} T^{20} + 104462653352 p^{5} T^{21} + 13921663582 p^{6} T^{22} + 1689153318 p^{7} T^{23} + 210198493 p^{8} T^{24} + 23295398 p^{9} T^{25} + 2682534 p^{10} T^{26} + 261025 p^{11} T^{27} + 27055 p^{12} T^{28} + 2226 p^{13} T^{29} + 213 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 7 T + 280 T^{2} + 1941 T^{3} + 36819 T^{4} + 236723 T^{5} + 3001538 T^{6} + 16733085 T^{7} + 165369035 T^{8} + 722769475 T^{9} + 6000489869 T^{10} + 13527964722 T^{11} + 96083212813 T^{12} - 631486751266 T^{13} - 3979227070975 T^{14} - 67297683048261 T^{15} - 328242403752856 T^{16} - 67297683048261 p T^{17} - 3979227070975 p^{2} T^{18} - 631486751266 p^{3} T^{19} + 96083212813 p^{4} T^{20} + 13527964722 p^{5} T^{21} + 6000489869 p^{6} T^{22} + 722769475 p^{7} T^{23} + 165369035 p^{8} T^{24} + 16733085 p^{9} T^{25} + 3001538 p^{10} T^{26} + 236723 p^{11} T^{27} + 36819 p^{12} T^{28} + 1941 p^{13} T^{29} + 280 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 17 T + 629 T^{2} - 8697 T^{3} + 182290 T^{4} - 2137083 T^{5} + 32981459 T^{6} - 336908436 T^{7} + 4229769743 T^{8} - 38357994156 T^{9} + 412089048578 T^{10} - 3360131192127 T^{11} + 31802161574506 T^{12} - 235036225203819 T^{13} + 1994209290677718 T^{14} - 13410404699960575 T^{15} + 103030599758100840 T^{16} - 13410404699960575 p T^{17} + 1994209290677718 p^{2} T^{18} - 235036225203819 p^{3} T^{19} + 31802161574506 p^{4} T^{20} - 3360131192127 p^{5} T^{21} + 412089048578 p^{6} T^{22} - 38357994156 p^{7} T^{23} + 4229769743 p^{8} T^{24} - 336908436 p^{9} T^{25} + 32981459 p^{10} T^{26} - 2137083 p^{11} T^{27} + 182290 p^{12} T^{28} - 8697 p^{13} T^{29} + 629 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 36 T + 913 T^{2} + 17113 T^{3} + 273000 T^{4} + 3797961 T^{5} + 48030646 T^{6} + 559307132 T^{7} + 6086072311 T^{8} + 62301564302 T^{9} + 603767934710 T^{10} + 5563808699135 T^{11} + 48921478505296 T^{12} + 411558232788555 T^{13} + 3318250510963467 T^{14} + 25672798469365790 T^{15} + 190714644740855808 T^{16} + 25672798469365790 p T^{17} + 3318250510963467 p^{2} T^{18} + 411558232788555 p^{3} T^{19} + 48921478505296 p^{4} T^{20} + 5563808699135 p^{5} T^{21} + 603767934710 p^{6} T^{22} + 62301564302 p^{7} T^{23} + 6086072311 p^{8} T^{24} + 559307132 p^{9} T^{25} + 48030646 p^{10} T^{26} + 3797961 p^{11} T^{27} + 273000 p^{12} T^{28} + 17113 p^{13} T^{29} + 913 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 53 T + 2054 T^{2} - 58020 T^{3} + 1378208 T^{4} - 27787940 T^{5} + 496055541 T^{6} - 7897503110 T^{7} + 114208673280 T^{8} - 1507410007798 T^{9} + 18334736549823 T^{10} - 206108939702655 T^{11} + 2153377265525746 T^{12} - 20941741366850577 T^{13} + 190198271160677726 T^{14} - 1613914978078759911 T^{15} + 12817854684594220410 T^{16} - 1613914978078759911 p T^{17} + 190198271160677726 p^{2} T^{18} - 20941741366850577 p^{3} T^{19} + 2153377265525746 p^{4} T^{20} - 206108939702655 p^{5} T^{21} + 18334736549823 p^{6} T^{22} - 1507410007798 p^{7} T^{23} + 114208673280 p^{8} T^{24} - 7897503110 p^{9} T^{25} + 496055541 p^{10} T^{26} - 27787940 p^{11} T^{27} + 1378208 p^{12} T^{28} - 58020 p^{13} T^{29} + 2054 p^{14} T^{30} - 53 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 34 T + 1064 T^{2} - 22434 T^{3} + 437036 T^{4} - 7030108 T^{5} + 106121038 T^{6} - 1416632082 T^{7} + 17942667751 T^{8} - 207431578622 T^{9} + 2291471070544 T^{10} - 23495219613202 T^{11} + 231222109393272 T^{12} - 2131632951073428 T^{13} + 18910901169635386 T^{14} - 2588559142754162 p T^{15} + 1270283921326043720 T^{16} - 2588559142754162 p^{2} T^{17} + 18910901169635386 p^{2} T^{18} - 2131632951073428 p^{3} T^{19} + 231222109393272 p^{4} T^{20} - 23495219613202 p^{5} T^{21} + 2291471070544 p^{6} T^{22} - 207431578622 p^{7} T^{23} + 17942667751 p^{8} T^{24} - 1416632082 p^{9} T^{25} + 106121038 p^{10} T^{26} - 7030108 p^{11} T^{27} + 437036 p^{12} T^{28} - 22434 p^{13} T^{29} + 1064 p^{14} T^{30} - 34 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 13 T + 520 T^{2} + 5916 T^{3} + 134362 T^{4} + 1370762 T^{5} + 23367066 T^{6} + 3267556 p T^{7} + 3125839920 T^{8} + 27311066850 T^{9} + 344573040849 T^{10} + 2823992642027 T^{11} + 32363829954948 T^{12} + 249098938160095 T^{13} + 2635621717292117 T^{14} + 19058378205421269 T^{15} + 188182292118403490 T^{16} + 19058378205421269 p T^{17} + 2635621717292117 p^{2} T^{18} + 249098938160095 p^{3} T^{19} + 32363829954948 p^{4} T^{20} + 2823992642027 p^{5} T^{21} + 344573040849 p^{6} T^{22} + 27311066850 p^{7} T^{23} + 3125839920 p^{8} T^{24} + 3267556 p^{10} T^{25} + 23367066 p^{10} T^{26} + 1370762 p^{11} T^{27} + 134362 p^{12} T^{28} + 5916 p^{13} T^{29} + 520 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 11 T + 577 T^{2} + 5476 T^{3} + 161619 T^{4} + 1340476 T^{5} + 29387300 T^{6} + 211414167 T^{7} + 3895183917 T^{8} + 23807666587 T^{9} + 402235710730 T^{10} + 2030440553164 T^{11} + 34236666394041 T^{12} + 140431687261364 T^{13} + 2570514515916833 T^{14} + 9011140612374591 T^{15} + 183634764282070924 T^{16} + 9011140612374591 p T^{17} + 2570514515916833 p^{2} T^{18} + 140431687261364 p^{3} T^{19} + 34236666394041 p^{4} T^{20} + 2030440553164 p^{5} T^{21} + 402235710730 p^{6} T^{22} + 23807666587 p^{7} T^{23} + 3895183917 p^{8} T^{24} + 211414167 p^{9} T^{25} + 29387300 p^{10} T^{26} + 1340476 p^{11} T^{27} + 161619 p^{12} T^{28} + 5476 p^{13} T^{29} + 577 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 34 T + 1115 T^{2} - 23154 T^{3} + 460744 T^{4} - 7187367 T^{5} + 108935312 T^{6} - 1394268664 T^{7} + 17618767756 T^{8} - 195105632167 T^{9} + 2169783578058 T^{10} - 21614430842848 T^{11} + 219581105770134 T^{12} - 2022959014643051 T^{13} + 19198283810395707 T^{14} - 166050376688534035 T^{15} + 1485387942558641898 T^{16} - 166050376688534035 p T^{17} + 19198283810395707 p^{2} T^{18} - 2022959014643051 p^{3} T^{19} + 219581105770134 p^{4} T^{20} - 21614430842848 p^{5} T^{21} + 2169783578058 p^{6} T^{22} - 195105632167 p^{7} T^{23} + 17618767756 p^{8} T^{24} - 1394268664 p^{9} T^{25} + 108935312 p^{10} T^{26} - 7187367 p^{11} T^{27} + 460744 p^{12} T^{28} - 23154 p^{13} T^{29} + 1115 p^{14} T^{30} - 34 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 7 T + 382 T^{2} + 3668 T^{3} + 94719 T^{4} + 967539 T^{5} + 17854563 T^{6} + 181056978 T^{7} + 2728549543 T^{8} + 26731147072 T^{9} + 351450625421 T^{10} + 3275017783243 T^{11} + 38868579674241 T^{12} + 342903690599056 T^{13} + 3743183139563186 T^{14} + 31064052433421241 T^{15} + 315747911179727280 T^{16} + 31064052433421241 p T^{17} + 3743183139563186 p^{2} T^{18} + 342903690599056 p^{3} T^{19} + 38868579674241 p^{4} T^{20} + 3275017783243 p^{5} T^{21} + 351450625421 p^{6} T^{22} + 26731147072 p^{7} T^{23} + 2728549543 p^{8} T^{24} + 181056978 p^{9} T^{25} + 17854563 p^{10} T^{26} + 967539 p^{11} T^{27} + 94719 p^{12} T^{28} + 3668 p^{13} T^{29} + 382 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 28 T + 815 T^{2} + 16636 T^{3} + 319549 T^{4} + 5231208 T^{5} + 80746833 T^{6} + 1126079574 T^{7} + 14916960165 T^{8} + 183779190107 T^{9} + 2164081308833 T^{10} + 24110537864110 T^{11} + 3114127063601 p T^{12} + 2649790079773340 T^{13} + 26287566885441775 T^{14} + 251123654701098791 T^{15} + 2328388197427034452 T^{16} + 251123654701098791 p T^{17} + 26287566885441775 p^{2} T^{18} + 2649790079773340 p^{3} T^{19} + 3114127063601 p^{5} T^{20} + 24110537864110 p^{5} T^{21} + 2164081308833 p^{6} T^{22} + 183779190107 p^{7} T^{23} + 14916960165 p^{8} T^{24} + 1126079574 p^{9} T^{25} + 80746833 p^{10} T^{26} + 5231208 p^{11} T^{27} + 319549 p^{12} T^{28} + 16636 p^{13} T^{29} + 815 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 8 T + 817 T^{2} + 5913 T^{3} + 333755 T^{4} + 2252561 T^{5} + 91045055 T^{6} + 583860801 T^{7} + 18610819114 T^{8} + 114391311396 T^{9} + 3027412974029 T^{10} + 17843073352680 T^{11} + 406009662544725 T^{12} + 2279656909874034 T^{13} + 45858418947650979 T^{14} + 242258138686473751 T^{15} + 4413352207547873354 T^{16} + 242258138686473751 p T^{17} + 45858418947650979 p^{2} T^{18} + 2279656909874034 p^{3} T^{19} + 406009662544725 p^{4} T^{20} + 17843073352680 p^{5} T^{21} + 3027412974029 p^{6} T^{22} + 114391311396 p^{7} T^{23} + 18610819114 p^{8} T^{24} + 583860801 p^{9} T^{25} + 91045055 p^{10} T^{26} + 2252561 p^{11} T^{27} + 333755 p^{12} T^{28} + 5913 p^{13} T^{29} + 817 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 4 T + 845 T^{2} - 2881 T^{3} + 362077 T^{4} - 1097779 T^{5} + 104658263 T^{6} - 293030355 T^{7} + 22875323071 T^{8} - 60674943102 T^{9} + 4008910226563 T^{10} - 10172073748165 T^{11} + 582319951392153 T^{12} - 1408663719038425 T^{13} + 71476501086101977 T^{14} - 162854465245286297 T^{15} + 7490902164935780884 T^{16} - 162854465245286297 p T^{17} + 71476501086101977 p^{2} T^{18} - 1408663719038425 p^{3} T^{19} + 582319951392153 p^{4} T^{20} - 10172073748165 p^{5} T^{21} + 4008910226563 p^{6} T^{22} - 60674943102 p^{7} T^{23} + 22875323071 p^{8} T^{24} - 293030355 p^{9} T^{25} + 104658263 p^{10} T^{26} - 1097779 p^{11} T^{27} + 362077 p^{12} T^{28} - 2881 p^{13} T^{29} + 845 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−1.76876574782501586437808203706, −1.61056418456049944682321235807, −1.50035232396133220811199233103, −1.47280953261788454780181377424, −1.47114998191284908478641958958, −1.45404320116941013218896171982, −1.41449486440582571734141478384, −1.39940051056200773077782018416, −1.32558719887657900315138682056, −1.16556344421679427988011756421, −0.931741627481619710012024255848, −0.894632711690210797986866354239, −0.76580510843631205474300549452, −0.69194733333475171402580850668, −0.63875661675144892506754511250, −0.53400594551124370509900842788, −0.47259469626452903222535558594, −0.41021904730105604163777015096, −0.40516943083758126123891164995, −0.39482895639102895654281415338, −0.39266143839998015510239297023, −0.38324712578409267005413829400, −0.38026660643540086042980123387, −0.23141376447649094822619378464, −0.02174827262580131625005937040, 0.02174827262580131625005937040, 0.23141376447649094822619378464, 0.38026660643540086042980123387, 0.38324712578409267005413829400, 0.39266143839998015510239297023, 0.39482895639102895654281415338, 0.40516943083758126123891164995, 0.41021904730105604163777015096, 0.47259469626452903222535558594, 0.53400594551124370509900842788, 0.63875661675144892506754511250, 0.69194733333475171402580850668, 0.76580510843631205474300549452, 0.894632711690210797986866354239, 0.931741627481619710012024255848, 1.16556344421679427988011756421, 1.32558719887657900315138682056, 1.39940051056200773077782018416, 1.41449486440582571734141478384, 1.45404320116941013218896171982, 1.47114998191284908478641958958, 1.47280953261788454780181377424, 1.50035232396133220811199233103, 1.61056418456049944682321235807, 1.76876574782501586437808203706
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.