| L(s) = 1 | − 10·4-s + 336·13-s + 13·16-s − 288·19-s − 424·25-s + 120·31-s + 592·37-s − 1.87e3·43-s − 3.36e3·52-s + 2.40e3·61-s − 300·64-s + 1.82e3·73-s + 2.88e3·76-s + 2.36e3·79-s + 5.71e3·97-s + 4.24e3·100-s − 5.40e3·103-s − 1.65e3·109-s − 5.78e3·121-s − 1.20e3·124-s + 127-s + 131-s + 137-s + 139-s − 5.92e3·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 5/4·4-s + 7.16·13-s + 0.203·16-s − 3.47·19-s − 3.39·25-s + 0.695·31-s + 2.63·37-s − 6.63·43-s − 8.96·52-s + 5.03·61-s − 0.585·64-s + 2.92·73-s + 4.34·76-s + 3.37·79-s + 5.97·97-s + 4.23·100-s − 5.16·103-s − 1.45·109-s − 4.34·121-s − 0.869·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.28·148-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.714485622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.714485622\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5 p T^{2} + 87 T^{4} + 65 p^{4} T^{6} + 167 p^{6} T^{8} + 65 p^{10} T^{10} + 87 p^{12} T^{12} + 5 p^{19} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 + 424 T^{2} + 113862 T^{4} + 21538016 T^{6} + 3080570771 T^{8} + 21538016 p^{6} T^{10} + 113862 p^{12} T^{12} + 424 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 + 5784 T^{2} + 14142406 T^{4} + 20701768608 T^{6} + 25999720964691 T^{8} + 20701768608 p^{6} T^{10} + 14142406 p^{12} T^{12} + 5784 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( ( 1 - 168 T + 17116 T^{2} - 1207896 T^{3} + 64621222 T^{4} - 1207896 p^{3} T^{5} + 17116 p^{6} T^{6} - 168 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 17 | \( 1 + 4320 T^{2} + 90997492 T^{4} + 307024797600 T^{6} + 3233455864654758 T^{8} + 307024797600 p^{6} T^{10} + 90997492 p^{12} T^{12} + 4320 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 + 144 T + 14886 T^{2} + 829440 T^{3} + 56352755 T^{4} + 829440 p^{3} T^{5} + 14886 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 + 37768 T^{2} + 966746070 T^{4} + 17415830892128 T^{6} + 242639055368418659 T^{8} + 17415830892128 p^{6} T^{10} + 966746070 p^{12} T^{12} + 37768 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( 1 + 75064 T^{2} + 4197383004 T^{4} + 150911401826504 T^{6} + 4338189011053329830 T^{8} + 150911401826504 p^{6} T^{10} + 4197383004 p^{12} T^{12} + 75064 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 - 60 T + 72250 T^{2} - 261480 p T^{3} + 80919757 p T^{4} - 261480 p^{4} T^{5} + 72250 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 p T + 44614 T^{2} + 138976 p T^{3} - 2636743133 T^{4} + 138976 p^{4} T^{5} + 44614 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( 1 + 388344 T^{2} + 69574817686 T^{4} + 7799366016841248 T^{6} + \)\(62\!\cdots\!31\)\( T^{8} + 7799366016841248 p^{6} T^{10} + 69574817686 p^{12} T^{12} + 388344 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 + 936 T + 590184 T^{2} + 242030232 T^{3} + 79565108834 T^{4} + 242030232 p^{3} T^{5} + 590184 p^{6} T^{6} + 936 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 + 539192 T^{2} + 147167990020 T^{4} + 25820308461522152 T^{6} + \)\(31\!\cdots\!14\)\( T^{8} + 25820308461522152 p^{6} T^{10} + 147167990020 p^{12} T^{12} + 539192 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 + 564888 T^{2} + 122098902844 T^{4} + 11434901409644904 T^{6} + \)\(73\!\cdots\!66\)\( T^{8} + 11434901409644904 p^{6} T^{10} + 122098902844 p^{12} T^{12} + 564888 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 + 502456 T^{2} + 216666468900 T^{4} + 63882710147039144 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} + 63882710147039144 p^{6} T^{10} + 216666468900 p^{12} T^{12} + 502456 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 - 1200 T + 1249988 T^{2} - 843411120 T^{3} + 464858253798 T^{4} - 843411120 p^{3} T^{5} + 1249988 p^{6} T^{6} - 1200 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 1023396 T^{2} - 24675840 T^{3} + 433884187622 T^{4} - 24675840 p^{3} T^{5} + 1023396 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( 1 + 1910344 T^{2} + 1810571288406 T^{4} + 1096791110591321888 T^{6} + \)\(46\!\cdots\!91\)\( T^{8} + 1096791110591321888 p^{6} T^{10} + 1810571288406 p^{12} T^{12} + 1910344 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( ( 1 - 912 T + 1041832 T^{2} - 557617680 T^{3} + 474743673106 T^{4} - 557617680 p^{3} T^{5} + 1041832 p^{6} T^{6} - 912 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 1184 T + 2396136 T^{2} - 1798718560 T^{3} + 1877500132370 T^{4} - 1798718560 p^{3} T^{5} + 2396136 p^{6} T^{6} - 1184 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 + 3779800 T^{2} + 6638776535772 T^{4} + 7032576780729679400 T^{6} + \)\(49\!\cdots\!98\)\( T^{8} + 7032576780729679400 p^{6} T^{10} + 6638776535772 p^{12} T^{12} + 3779800 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( 1 + 1909176 T^{2} + 3175905142966 T^{4} + 3100144053826523232 T^{6} + \)\(26\!\cdots\!11\)\( T^{8} + 3100144053826523232 p^{6} T^{10} + 3175905142966 p^{12} T^{12} + 1909176 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( ( 1 - 2856 T + 5804884 T^{2} - 7716365688 T^{3} + 8573207370022 T^{4} - 7716365688 p^{3} T^{5} + 5804884 p^{6} T^{6} - 2856 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.85801123059411121913492777103, −3.67180858392766680597733582261, −3.49611404819188545812844493654, −3.48045965269665244402197762674, −3.37312515023666551480724375554, −3.04558551265308480324268362756, −2.86773863087654842400470117102, −2.80673665770956677217644193191, −2.70888079683286809722394627717, −2.39198215531027965805993533807, −2.23286718895847929345462738290, −2.13612829115830924689435886502, −2.11059585158939446179143385588, −1.69223889413001091209951861786, −1.64727491552508987929288088891, −1.56985479462041078611502291291, −1.55240959726617889010219230939, −1.38860969684012377389721046098, −1.19437958284839200176063101959, −0.907769866796201208928267673463, −0.63273476804702374096450144198, −0.58326202738954682314329749742, −0.49965696433510390071892428664, −0.47755916501800836198591919688, −0.07471588277330571986434297865,
0.07471588277330571986434297865, 0.47755916501800836198591919688, 0.49965696433510390071892428664, 0.58326202738954682314329749742, 0.63273476804702374096450144198, 0.907769866796201208928267673463, 1.19437958284839200176063101959, 1.38860969684012377389721046098, 1.55240959726617889010219230939, 1.56985479462041078611502291291, 1.64727491552508987929288088891, 1.69223889413001091209951861786, 2.11059585158939446179143385588, 2.13612829115830924689435886502, 2.23286718895847929345462738290, 2.39198215531027965805993533807, 2.70888079683286809722394627717, 2.80673665770956677217644193191, 2.86773863087654842400470117102, 3.04558551265308480324268362756, 3.37312515023666551480724375554, 3.48045965269665244402197762674, 3.49611404819188545812844493654, 3.67180858392766680597733582261, 3.85801123059411121913492777103
Plot not available for L-functions of degree greater than 10.
