L(s) = 1 | − 3·2-s − 7·3-s + 4-s − 4·5-s + 21·6-s − 9·7-s + 4·8-s + 28·9-s + 12·10-s − 4·11-s − 7·12-s − 4·13-s + 27·14-s + 28·15-s − 2·16-s − 15·17-s − 84·18-s − 2·19-s − 4·20-s + 63·21-s + 12·22-s − 22·23-s − 28·24-s − 6·25-s + 12·26-s − 84·27-s − 9·28-s + ⋯ |
L(s) = 1 | − 2.12·2-s − 4.04·3-s + 1/2·4-s − 1.78·5-s + 8.57·6-s − 3.40·7-s + 1.41·8-s + 28/3·9-s + 3.79·10-s − 1.20·11-s − 2.02·12-s − 1.10·13-s + 7.21·14-s + 7.22·15-s − 1/2·16-s − 3.63·17-s − 19.7·18-s − 0.458·19-s − 0.894·20-s + 13.7·21-s + 2.55·22-s − 4.58·23-s − 5.71·24-s − 6/5·25-s + 2.35·26-s − 16.1·27-s − 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 139^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 139^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T )^{7} \) |
| 139 | \( ( 1 + T )^{7} \) |
good | 2 | \( 1 + 3 T + p^{3} T^{2} + 17 T^{3} + 33 T^{4} + 29 p T^{5} + 47 p T^{6} + 17 p^{3} T^{7} + 47 p^{2} T^{8} + 29 p^{3} T^{9} + 33 p^{3} T^{10} + 17 p^{4} T^{11} + p^{8} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 4 T + 22 T^{2} + 13 p T^{3} + 49 p T^{4} + 608 T^{5} + 1791 T^{6} + 3681 T^{7} + 1791 p T^{8} + 608 p^{2} T^{9} + 49 p^{4} T^{10} + 13 p^{5} T^{11} + 22 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + 9 T + 58 T^{2} + 269 T^{3} + 1068 T^{4} + 3568 T^{5} + 10919 T^{6} + 29889 T^{7} + 10919 p T^{8} + 3568 p^{2} T^{9} + 1068 p^{3} T^{10} + 269 p^{4} T^{11} + 58 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 + 4 T + 32 T^{2} + 57 T^{3} + 413 T^{4} + 646 T^{5} + 6467 T^{6} + 11895 T^{7} + 6467 p T^{8} + 646 p^{2} T^{9} + 413 p^{3} T^{10} + 57 p^{4} T^{11} + 32 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 13 | \( 1 + 4 T + 29 T^{2} + 61 T^{3} + 424 T^{4} + 937 T^{5} + 7760 T^{6} + 17452 T^{7} + 7760 p T^{8} + 937 p^{2} T^{9} + 424 p^{3} T^{10} + 61 p^{4} T^{11} + 29 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 17 | \( 1 + 15 T + 144 T^{2} + 984 T^{3} + 5934 T^{4} + 31481 T^{5} + 155217 T^{6} + 671424 T^{7} + 155217 p T^{8} + 31481 p^{2} T^{9} + 5934 p^{3} T^{10} + 984 p^{4} T^{11} + 144 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) |
| 19 | \( 1 + 2 T + 72 T^{2} + 130 T^{3} + 2610 T^{4} + 3878 T^{5} + 66743 T^{6} + 80908 T^{7} + 66743 p T^{8} + 3878 p^{2} T^{9} + 2610 p^{3} T^{10} + 130 p^{4} T^{11} + 72 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 23 | \( 1 + 22 T + 308 T^{2} + 3162 T^{3} + 26226 T^{4} + 181770 T^{5} + 1080231 T^{6} + 5556044 T^{7} + 1080231 p T^{8} + 181770 p^{2} T^{9} + 26226 p^{3} T^{10} + 3162 p^{4} T^{11} + 308 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) |
| 29 | \( 1 + 12 T + 157 T^{2} + 1275 T^{3} + 10054 T^{4} + 62571 T^{5} + 393372 T^{6} + 2068176 T^{7} + 393372 p T^{8} + 62571 p^{2} T^{9} + 10054 p^{3} T^{10} + 1275 p^{4} T^{11} + 157 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) |
| 31 | \( 1 + 9 T + 190 T^{2} + 1419 T^{3} + 15914 T^{4} + 99660 T^{5} + 778295 T^{6} + 3989155 T^{7} + 778295 p T^{8} + 99660 p^{2} T^{9} + 15914 p^{3} T^{10} + 1419 p^{4} T^{11} + 190 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) |
| 37 | \( 1 - 4 T + 56 T^{2} - 303 T^{3} + 1972 T^{4} - 1109 T^{5} + 33059 T^{6} + 136208 T^{7} + 33059 p T^{8} - 1109 p^{2} T^{9} + 1972 p^{3} T^{10} - 303 p^{4} T^{11} + 56 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 41 | \( 1 + 5 T + 233 T^{2} + 812 T^{3} + 23252 T^{4} + 56636 T^{5} + 1383846 T^{6} + 2603422 T^{7} + 1383846 p T^{8} + 56636 p^{2} T^{9} + 23252 p^{3} T^{10} + 812 p^{4} T^{11} + 233 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) |
| 43 | \( 1 + 11 T + 160 T^{2} + 1170 T^{3} + 13438 T^{4} + 89229 T^{5} + 814115 T^{6} + 4523516 T^{7} + 814115 p T^{8} + 89229 p^{2} T^{9} + 13438 p^{3} T^{10} + 1170 p^{4} T^{11} + 160 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) |
| 47 | \( 1 + 38 T + 888 T^{2} + 14689 T^{3} + 191474 T^{4} + 2025787 T^{5} + 17924059 T^{6} + 133418588 T^{7} + 17924059 p T^{8} + 2025787 p^{2} T^{9} + 191474 p^{3} T^{10} + 14689 p^{4} T^{11} + 888 p^{5} T^{12} + 38 p^{6} T^{13} + p^{7} T^{14} \) |
| 53 | \( 1 + 20 T + 479 T^{2} + 6264 T^{3} + 85849 T^{4} + 820940 T^{5} + 7967647 T^{6} + 57785616 T^{7} + 7967647 p T^{8} + 820940 p^{2} T^{9} + 85849 p^{3} T^{10} + 6264 p^{4} T^{11} + 479 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) |
| 59 | \( 1 + T + 250 T^{2} + 706 T^{3} + 28624 T^{4} + 128767 T^{5} + 2134179 T^{6} + 10718860 T^{7} + 2134179 p T^{8} + 128767 p^{2} T^{9} + 28624 p^{3} T^{10} + 706 p^{4} T^{11} + 250 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 61 | \( 1 - 15 T + 116 T^{2} - 310 T^{3} - 642 T^{4} + 33487 T^{5} - 1835 p T^{6} + 432828 T^{7} - 1835 p^{2} T^{8} + 33487 p^{2} T^{9} - 642 p^{3} T^{10} - 310 p^{4} T^{11} + 116 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) |
| 67 | \( 1 + 11 T + 334 T^{2} + 2799 T^{3} + 51664 T^{4} + 363260 T^{5} + 5143579 T^{6} + 30261919 T^{7} + 5143579 p T^{8} + 363260 p^{2} T^{9} + 51664 p^{3} T^{10} + 2799 p^{4} T^{11} + 334 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) |
| 71 | \( 1 + 14 T + 459 T^{2} + 4911 T^{3} + 90310 T^{4} + 768107 T^{5} + 10182936 T^{6} + 69582980 T^{7} + 10182936 p T^{8} + 768107 p^{2} T^{9} + 90310 p^{3} T^{10} + 4911 p^{4} T^{11} + 459 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) |
| 73 | \( 1 + 12 T + 414 T^{2} + 3876 T^{3} + 77044 T^{4} + 588316 T^{5} + 8617653 T^{6} + 53894920 T^{7} + 8617653 p T^{8} + 588316 p^{2} T^{9} + 77044 p^{3} T^{10} + 3876 p^{4} T^{11} + 414 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) |
| 79 | \( 1 + 6 T + 104 T^{2} + 579 T^{3} + 16863 T^{4} + 80724 T^{5} + 1528263 T^{6} + 7838587 T^{7} + 1528263 p T^{8} + 80724 p^{2} T^{9} + 16863 p^{3} T^{10} + 579 p^{4} T^{11} + 104 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) |
| 83 | \( 1 + 32 T + 832 T^{2} + 15511 T^{3} + 241815 T^{4} + 3156984 T^{5} + 35588003 T^{6} + 347001055 T^{7} + 35588003 p T^{8} + 3156984 p^{2} T^{9} + 241815 p^{3} T^{10} + 15511 p^{4} T^{11} + 832 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) |
| 89 | \( 1 + 9 T + 444 T^{2} + 2841 T^{3} + 85176 T^{4} + 398710 T^{5} + 10176081 T^{6} + 38717511 T^{7} + 10176081 p T^{8} + 398710 p^{2} T^{9} + 85176 p^{3} T^{10} + 2841 p^{4} T^{11} + 444 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) |
| 97 | \( 1 - 19 T + 482 T^{2} - 7596 T^{3} + 113076 T^{4} - 1388093 T^{5} + 16519353 T^{6} - 160876424 T^{7} + 16519353 p T^{8} - 1388093 p^{2} T^{9} + 113076 p^{3} T^{10} - 7596 p^{4} T^{11} + 482 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.96247082726623170263752002452, −5.89886068360771188123839335622, −5.83749055960690448144681570704, −5.82589520025088350177289337827, −5.46678811737387943808625028597, −5.14483594754423126172327518010, −5.10727159818873551993638480274, −4.81283671148886929023367953756, −4.71461977632874904236288325730, −4.66526790109080627696554983038, −4.61346061708368294969233045615, −4.17274964139591714806025707072, −4.07710324257157591677458531012, −3.86097209217015729485554764560, −3.84732683962295676749492920612, −3.65800060837805652788710585142, −3.64797406562986689306987567064, −3.10746443887799870045305889729, −3.07790418753237868946931671137, −2.74886067494718385619909387091, −2.40877896981626728238838178011, −1.89665412920588854636040235489, −1.87661946460175157985196812033, −1.72380942716762663811508730695, −1.53864928881560772176990949949, 0, 0, 0, 0, 0, 0, 0,
1.53864928881560772176990949949, 1.72380942716762663811508730695, 1.87661946460175157985196812033, 1.89665412920588854636040235489, 2.40877896981626728238838178011, 2.74886067494718385619909387091, 3.07790418753237868946931671137, 3.10746443887799870045305889729, 3.64797406562986689306987567064, 3.65800060837805652788710585142, 3.84732683962295676749492920612, 3.86097209217015729485554764560, 4.07710324257157591677458531012, 4.17274964139591714806025707072, 4.61346061708368294969233045615, 4.66526790109080627696554983038, 4.71461977632874904236288325730, 4.81283671148886929023367953756, 5.10727159818873551993638480274, 5.14483594754423126172327518010, 5.46678811737387943808625028597, 5.82589520025088350177289337827, 5.83749055960690448144681570704, 5.89886068360771188123839335622, 5.96247082726623170263752002452
Plot not available for L-functions of degree greater than 10.