| L(s) = 1 | + 2·2-s + 4-s + 4·7-s − 2·8-s + 4·13-s + 8·14-s − 4·16-s + 20·19-s + 8·26-s + 4·28-s − 4·31-s − 2·32-s + 16·37-s + 40·38-s + 18·41-s + 10·43-s + 12·47-s + 12·49-s + 4·52-s − 24·53-s − 8·56-s − 6·59-s − 16·61-s − 8·62-s + 3·64-s − 14·67-s − 24·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.707·8-s + 1.10·13-s + 2.13·14-s − 16-s + 4.58·19-s + 1.56·26-s + 0.755·28-s − 0.718·31-s − 0.353·32-s + 2.63·37-s + 6.48·38-s + 2.81·41-s + 1.52·43-s + 1.75·47-s + 12/7·49-s + 0.554·52-s − 3.29·53-s − 1.06·56-s − 0.781·59-s − 2.04·61-s − 1.01·62-s + 3/8·64-s − 1.71·67-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.91720404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.91720404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 8 T^{2} + 8 T^{3} + 199 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 190 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 144 T^{3} + 2259 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 18 p T^{5} - 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 848 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 95 T^{2} - 190 T^{3} + 4 T^{4} - 190 p T^{5} - 95 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71981517267625658596506015499, −6.50827236564612891639780671963, −6.07598111777561106519912966639, −6.07523406253942200591376850841, −5.96798187137750553835905045026, −5.67440497899875455296477728877, −5.65319169625061370369477086694, −5.29873464672879007993362095721, −4.98700738818076558701185602972, −4.71382872234271675805605402611, −4.70645263611890698509203204958, −4.45941557643296133835659790203, −4.16560234923717433500765229917, −4.11103475144399128030165521335, −3.50878746298821346101143924832, −3.34707267077599604037004505305, −3.32701819704112125281820913082, −2.82395994427861720506980348040, −2.80649797184267824219837101912, −2.47083233220812595710562016841, −1.82442962433535292796992759998, −1.70906613926827907626348455365, −1.16690437668049483212765498259, −0.856592559022417067197807360179, −0.817907983125819327648891589277,
0.817907983125819327648891589277, 0.856592559022417067197807360179, 1.16690437668049483212765498259, 1.70906613926827907626348455365, 1.82442962433535292796992759998, 2.47083233220812595710562016841, 2.80649797184267824219837101912, 2.82395994427861720506980348040, 3.32701819704112125281820913082, 3.34707267077599604037004505305, 3.50878746298821346101143924832, 4.11103475144399128030165521335, 4.16560234923717433500765229917, 4.45941557643296133835659790203, 4.70645263611890698509203204958, 4.71382872234271675805605402611, 4.98700738818076558701185602972, 5.29873464672879007993362095721, 5.65319169625061370369477086694, 5.67440497899875455296477728877, 5.96798187137750553835905045026, 6.07523406253942200591376850841, 6.07598111777561106519912966639, 6.50827236564612891639780671963, 6.71981517267625658596506015499
