| L(s) = 1 | − 4·4-s + 4·7-s − 20·13-s + 12·16-s − 28·19-s − 16·28-s + 8·31-s − 20·37-s − 152·43-s − 186·49-s + 80·52-s − 100·61-s − 32·64-s − 284·67-s − 164·73-s + 112·76-s − 76·79-s − 80·91-s − 20·97-s + 100·103-s + 248·109-s + 48·112-s + 124·121-s − 32·124-s + 127-s + 131-s − 112·133-s + ⋯ |
| L(s) = 1 | − 4-s + 4/7·7-s − 1.53·13-s + 3/4·16-s − 1.47·19-s − 4/7·28-s + 8/31·31-s − 0.540·37-s − 3.53·43-s − 3.79·49-s + 1.53·52-s − 1.63·61-s − 1/2·64-s − 4.23·67-s − 2.24·73-s + 1.47·76-s − 0.962·79-s − 0.879·91-s − 0.206·97-s + 0.970·103-s + 2.27·109-s + 3/7·112-s + 1.02·121-s − 0.258·124-s + 0.00787·127-s + 0.00763·131-s − 0.842·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05443470016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05443470016\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 10 T + 3 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 506 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 14 T + 411 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1180 T^{2} + 700422 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2860 T^{2} + 3407622 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 486 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 2403 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2764 T^{2} + 6265446 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 76 T + 3702 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3796 T^{2} + 8177766 T^{4} - 3796 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 788 T^{2} + 11737158 T^{4} + 788 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6890 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 50 T + 2307 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 71 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4108 T^{2} - 8461722 T^{4} - 4108 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 82 T + 11979 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 38 T + 3843 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2860 T^{2} - 27506298 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8500 T^{2} + 43184742 T^{4} - 8500 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 10 T + 9843 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.48838001017415160106281048664, −6.34257151668683396153770126537, −6.19544326571280078442013290352, −6.12558836843204864305127552457, −5.94053660008336157970209659731, −5.24448783777604828001469127332, −5.23112167618966475677213576044, −5.16239445012981527144868313832, −4.80433674921516827406947783085, −4.54877339705749199953912216189, −4.53924924191866326702669477752, −4.38556896450123690912621564985, −4.06327539989963875355909634137, −3.55284081082544322235273780167, −3.28615108796380470518834826152, −3.27195174244089798627128773724, −3.01110130333970664896872320489, −2.60436046834991785250172729912, −2.31454236646724439031814178995, −1.84825153544375738746849332578, −1.68983233527203345584697318247, −1.40796414604885275460770256946, −1.21193120330811819145762739719, −0.22337534680822890408647425378, −0.088612395393307361190970037159,
0.088612395393307361190970037159, 0.22337534680822890408647425378, 1.21193120330811819145762739719, 1.40796414604885275460770256946, 1.68983233527203345584697318247, 1.84825153544375738746849332578, 2.31454236646724439031814178995, 2.60436046834991785250172729912, 3.01110130333970664896872320489, 3.27195174244089798627128773724, 3.28615108796380470518834826152, 3.55284081082544322235273780167, 4.06327539989963875355909634137, 4.38556896450123690912621564985, 4.53924924191866326702669477752, 4.54877339705749199953912216189, 4.80433674921516827406947783085, 5.16239445012981527144868313832, 5.23112167618966475677213576044, 5.24448783777604828001469127332, 5.94053660008336157970209659731, 6.12558836843204864305127552457, 6.19544326571280078442013290352, 6.34257151668683396153770126537, 6.48838001017415160106281048664
