| L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s − 4·11-s + 4·13-s + 16·17-s − 8·19-s − 4·23-s + 25-s − 10·29-s + 4·31-s − 8·35-s − 6·41-s + 16·43-s − 6·45-s − 4·47-s + 15·49-s + 24·53-s − 8·55-s − 8·59-s + 10·61-s + 12·63-s + 8·65-s − 16·67-s + 24·71-s + 16·77-s − 24·79-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s − 1.20·11-s + 1.10·13-s + 3.88·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s − 1.35·35-s − 0.937·41-s + 2.43·43-s − 0.894·45-s − 0.583·47-s + 15/7·49-s + 3.29·53-s − 1.07·55-s − 1.04·59-s + 1.28·61-s + 1.51·63-s + 0.992·65-s − 1.95·67-s + 2.84·71-s + 1.82·77-s − 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611672430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611672430\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + T^{2} + 4 T^{3} + 64 T^{4} + 4 p T^{5} + p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 32 T^{3} - 53 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 4 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 234 T^{3} - 1308 T^{4} - 234 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 118 T^{2} - 832 T^{3} + 6187 T^{4} - 832 p T^{5} + 118 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 79 T^{2} + 4 T^{3} + 6064 T^{4} + 4 p T^{5} - 79 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 38 T^{2} - 736 T^{3} - 6581 T^{4} - 736 p T^{5} + 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 10 T - 35 T^{2} - 130 T^{3} + 8404 T^{4} - 130 p T^{5} - 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 61 T^{2} + 976 T^{3} + 16384 T^{4} + 976 p T^{5} + 61 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 24 T + 286 T^{2} + 3168 T^{3} + 32355 T^{4} + 3168 p T^{5} + 286 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 53 T^{2} - 592 T^{3} + 13072 T^{4} - 592 p T^{5} + 53 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 134 T^{2} + 176 T^{3} + 11539 T^{4} + 176 p T^{5} - 134 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−8.277912963523894784439272872381, −8.176424050990081386803865532050, −7.65649043858417910371252090327, −7.63688865967392468488734496618, −7.47103326138563014872651638181, −6.95366829851945055508569903432, −6.80345614137164171079695257944, −6.42315505890095790903619972942, −6.12528411835334544016819336051, −5.82399099140721470929319998351, −5.75293896292793653187458719451, −5.58765064207695574089481178726, −5.41967964398944480144057992240, −5.20002225972975219702325681800, −4.46795034579811914521682656604, −4.09945927414284025515760451864, −3.99224044203238754945912941126, −3.55245400186546745904907568799, −3.21904497362344097652353402901, −3.06226312921889317955751392042, −2.67454307753259937510516651636, −2.18073709214587891063452903354, −1.98143224667969022500237231600, −1.15790167370062726802460172380, −0.56731375051423729528979153887,
0.56731375051423729528979153887, 1.15790167370062726802460172380, 1.98143224667969022500237231600, 2.18073709214587891063452903354, 2.67454307753259937510516651636, 3.06226312921889317955751392042, 3.21904497362344097652353402901, 3.55245400186546745904907568799, 3.99224044203238754945912941126, 4.09945927414284025515760451864, 4.46795034579811914521682656604, 5.20002225972975219702325681800, 5.41967964398944480144057992240, 5.58765064207695574089481178726, 5.75293896292793653187458719451, 5.82399099140721470929319998351, 6.12528411835334544016819336051, 6.42315505890095790903619972942, 6.80345614137164171079695257944, 6.95366829851945055508569903432, 7.47103326138563014872651638181, 7.63688865967392468488734496618, 7.65649043858417910371252090327, 8.176424050990081386803865532050, 8.277912963523894784439272872381
