| L(s) = 1 | − 12·7-s − 20·13-s + 28·19-s + 96·25-s + 64·31-s − 72·37-s − 172·43-s − 100·49-s + 160·61-s + 172·67-s − 32·73-s − 196·79-s + 240·91-s + 128·97-s − 64·103-s − 20·109-s + 360·121-s + 127-s + 131-s − 336·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.71·7-s − 1.53·13-s + 1.47·19-s + 3.83·25-s + 2.06·31-s − 1.94·37-s − 4·43-s − 2.04·49-s + 2.62·61-s + 2.56·67-s − 0.438·73-s − 2.48·79-s + 2.63·91-s + 1.31·97-s − 0.621·103-s − 0.183·109-s + 2.97·121-s + 0.00787·127-s + 0.00763·131-s − 2.52·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03347206166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03347206166\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 3551 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 6 T + 104 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 360 T^{2} + 59954 T^{4} - 360 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 10 T + 27 p T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 792 T^{2} + 323495 T^{4} - 792 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 14 T + 624 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1728 T^{2} + 1268546 T^{4} - 1728 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3208 T^{2} + 3981303 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 32 T + 726 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 36 T + 179 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 952 T^{2} - 2112174 T^{4} - 952 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 p T + 5040 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2452 T^{2} + 11033910 T^{4} - 2452 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2712 T^{2} + 9939698 T^{4} - 2712 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7380 T^{2} + 27472022 T^{4} - 7380 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 80 T + 8175 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 86 T + 7944 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8832 T^{2} + 53728706 T^{4} - 8832 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 4647 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 98 T + 9840 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3732 T^{2} + 9617798 T^{4} - 3732 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11920 T^{2} + 77845407 T^{4} - 11920 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 64 T + 18390 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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
−6.81148496792708603927515515788, −6.57571572496677368082499743428, −6.33359928643722313546178149215, −6.23709477364852096706118421937, −5.80714237895704741431673168824, −5.50516454955359807219720168651, −5.19647445004999928217663068812, −5.10061880943703886359358345501, −4.93489880509814984738029732939, −4.86685595294955545378978299492, −4.47575173160329762685465292369, −4.35420119691295971545734555535, −3.77688787349482986857362047432, −3.49199790350364799042333159412, −3.23811484055528514636435586499, −3.20667834501204462250017985794, −3.05434675209512289447850207783, −2.80624719371296193127337296902, −2.27170508435971344302792200751, −2.25681620357254025839720024230, −1.51099803289185470825212507548, −1.50646980540989415143803627772, −0.78371458558442642977595126814, −0.77299436795749735690065465656, −0.02959558852751293296052229382,
0.02959558852751293296052229382, 0.77299436795749735690065465656, 0.78371458558442642977595126814, 1.50646980540989415143803627772, 1.51099803289185470825212507548, 2.25681620357254025839720024230, 2.27170508435971344302792200751, 2.80624719371296193127337296902, 3.05434675209512289447850207783, 3.20667834501204462250017985794, 3.23811484055528514636435586499, 3.49199790350364799042333159412, 3.77688787349482986857362047432, 4.35420119691295971545734555535, 4.47575173160329762685465292369, 4.86685595294955545378978299492, 4.93489880509814984738029732939, 5.10061880943703886359358345501, 5.19647445004999928217663068812, 5.50516454955359807219720168651, 5.80714237895704741431673168824, 6.23709477364852096706118421937, 6.33359928643722313546178149215, 6.57571572496677368082499743428, 6.81148496792708603927515515788
