| L(s) = 1 | − 6·2-s − 3-s + 23·4-s + 6·5-s + 6·6-s + 3·7-s − 72·8-s − 2·9-s − 36·10-s + 2·11-s − 23·12-s − 6·13-s − 18·14-s − 6·15-s + 199·16-s + 3·17-s + 12·18-s + 5·19-s + 138·20-s − 3·21-s − 12·22-s + 22·23-s + 72·24-s + 31·25-s + 36·26-s − 3·27-s + 69·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 0.577·3-s + 23/2·4-s + 2.68·5-s + 2.44·6-s + 1.13·7-s − 25.4·8-s − 2/3·9-s − 11.3·10-s + 0.603·11-s − 6.63·12-s − 1.66·13-s − 4.81·14-s − 1.54·15-s + 49.7·16-s + 0.727·17-s + 2.82·18-s + 1.14·19-s + 30.8·20-s − 0.654·21-s − 2.55·22-s + 4.58·23-s + 14.6·24-s + 31/5·25-s + 7.06·26-s − 0.577·27-s + 13.0·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.368930595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368930595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + T + p T^{2} + 8 T^{3} + 8 T^{4} - 7 T^{5} + 2 p T^{6} - 56 T^{7} - 137 T^{8} - 56 p T^{9} + 2 p^{3} T^{10} - 7 p^{3} T^{11} + 8 p^{4} T^{12} + 8 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} - 36 T^{3} + 108 T^{4} - 219 T^{5} + 122 p T^{6} - 2628 T^{7} + 5483 T^{8} - 2628 p T^{9} + 122 p^{3} T^{10} - 219 p^{3} T^{11} + 108 p^{4} T^{12} - 36 p^{5} T^{13} + p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 2 T - 9 T^{2} + 42 T^{3} - 124 T^{4} + 84 T^{5} + 629 T^{6} - 3734 T^{7} + 13467 T^{8} - 3734 p T^{9} + 629 p^{2} T^{10} + 84 p^{3} T^{11} - 124 p^{4} T^{12} + 42 p^{5} T^{13} - 9 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 6 T + p T^{2} - 102 T^{3} - 792 T^{4} - 2892 T^{5} - 1009 T^{6} + 36864 T^{7} + 214223 T^{8} + 36864 p T^{9} - 1009 p^{2} T^{10} - 2892 p^{3} T^{11} - 792 p^{4} T^{12} - 102 p^{5} T^{13} + p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 - 3 T + 7 T^{2} + 144 T^{3} - 732 T^{4} + 1641 T^{5} + 1934 T^{6} - 49698 T^{7} + 169253 T^{8} - 49698 p T^{9} + 1934 p^{2} T^{10} + 1641 p^{3} T^{11} - 732 p^{4} T^{12} + 144 p^{5} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 5 T + p T^{2} - 160 T^{3} + 800 T^{4} - 2765 T^{5} + 914 p T^{6} - 88480 T^{7} + 312079 T^{8} - 88480 p T^{9} + 914 p^{3} T^{10} - 2765 p^{3} T^{11} + 800 p^{4} T^{12} - 160 p^{5} T^{13} + p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 11 T + 38 T^{2} - 125 T^{3} + 821 T^{4} - 125 p T^{5} + 38 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T - 57 T^{2} + 4 T^{3} + 3368 T^{4} + 4 p T^{5} - 57 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} - 528 T^{3} - 6224 T^{4} - 39696 T^{5} - 2521 T^{6} + 1579436 T^{7} + 15159727 T^{8} + 1579436 p T^{9} - 2521 p^{2} T^{10} - 39696 p^{3} T^{11} - 6224 p^{4} T^{12} - 528 p^{5} T^{13} + p^{7} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + T + 28 T^{2} + 443 T^{3} - 232 T^{4} - 6152 T^{5} - 2224 T^{6} - 797146 T^{7} - 5551967 T^{8} - 797146 p T^{9} - 2224 p^{2} T^{10} - 6152 p^{3} T^{11} - 232 p^{4} T^{12} + 443 p^{5} T^{13} + 28 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 385 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 21 T + 323 T^{2} + 3588 T^{3} + 35568 T^{4} + 324633 T^{5} + 2767366 T^{6} + 22572954 T^{7} + 167761313 T^{8} + 22572954 p T^{9} + 2767366 p^{2} T^{10} + 324633 p^{3} T^{11} + 35568 p^{4} T^{12} + 3588 p^{5} T^{13} + 323 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 5 T - T^{2} - 900 T^{3} - 8060 T^{4} - 17835 T^{5} + 134126 T^{6} + 3072320 T^{7} + 18108079 T^{8} + 3072320 p T^{9} + 134126 p^{2} T^{10} - 17835 p^{3} T^{11} - 8060 p^{4} T^{12} - 900 p^{5} T^{13} - p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 4 T - 117 T^{2} + 4 T^{3} + 12128 T^{4} + 4 p T^{5} - 117 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 7 T - 4 T^{2} + 1227 T^{3} - 13364 T^{4} + 49044 T^{5} + 214904 T^{6} - 6815734 T^{7} + 62113357 T^{8} - 6815734 p T^{9} + 214904 p^{2} T^{10} + 49044 p^{3} T^{11} - 13364 p^{4} T^{12} + 1227 p^{5} T^{13} - 4 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 + 21 T + 208 T^{2} - 297 T^{3} - 27162 T^{4} - 338082 T^{5} - 1041544 T^{6} + 15918264 T^{7} + 260450783 T^{8} + 15918264 p T^{9} - 1041544 p^{2} T^{10} - 338082 p^{3} T^{11} - 27162 p^{4} T^{12} - 297 p^{5} T^{13} + 208 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 14 T + 183 T^{2} - 42 T^{3} - 7462 T^{4} + 179508 T^{5} + 16211 T^{6} - 9573326 T^{7} + 219347043 T^{8} - 9573326 p T^{9} + 16211 p^{2} T^{10} + 179508 p^{3} T^{11} - 7462 p^{4} T^{12} - 42 p^{5} T^{13} + 183 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 5 T - 29 T^{2} + 785 T^{3} - 624 T^{4} + 785 p T^{5} - 29 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 345 p T^{5} + 182 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.47045329453765318506814609737, −4.13750987003398233053581278421, −4.09782730757576199477479132867, −3.76159487124326704021399026823, −3.39177766940073419419721899966, −3.30815963737098415541682036502, −3.13743565435312915766343593389, −3.08583520114776822871969633819, −3.08305451208115529839210966350, −2.91417491809159066820847584683, −2.81214609046502376923362719476, −2.72562334396295455238466597356, −2.72329667748480717692769130805, −2.32322323920619696404217915511, −2.00147103659247028781458586740, −1.99563207359567482999859754532, −1.98482014245594317101545805412, −1.70711799803223450735004374021, −1.56426143540442415862744340013, −1.36707271579727279434767534580, −0.995672606876184435381753046402, −0.983601736695836751077992218048, −0.983392101654024871022079181097, −0.53757221368283849994529731969, −0.53685880572123514149093412181,
0.53685880572123514149093412181, 0.53757221368283849994529731969, 0.983392101654024871022079181097, 0.983601736695836751077992218048, 0.995672606876184435381753046402, 1.36707271579727279434767534580, 1.56426143540442415862744340013, 1.70711799803223450735004374021, 1.98482014245594317101545805412, 1.99563207359567482999859754532, 2.00147103659247028781458586740, 2.32322323920619696404217915511, 2.72329667748480717692769130805, 2.72562334396295455238466597356, 2.81214609046502376923362719476, 2.91417491809159066820847584683, 3.08305451208115529839210966350, 3.08583520114776822871969633819, 3.13743565435312915766343593389, 3.30815963737098415541682036502, 3.39177766940073419419721899966, 3.76159487124326704021399026823, 4.09782730757576199477479132867, 4.13750987003398233053581278421, 4.47045329453765318506814609737
Plot not available for L-functions of degree greater than 10.
