| L(s) = 1 | − 2·5-s − 9-s + 6·13-s + 25-s − 6·29-s + 2·37-s + 2·45-s + 2·49-s + 2·53-s − 2·61-s − 12·65-s − 2·73-s + 81-s + 4·97-s − 8·101-s + 4·113-s − 6·117-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·5-s − 9-s + 6·13-s + 25-s − 6·29-s + 2·37-s + 2·45-s + 2·49-s + 2·53-s − 2·61-s − 12·65-s − 2·73-s + 81-s + 4·97-s − 8·101-s + 4·113-s − 6·117-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4275957143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4275957143\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| good | 3 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 7 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 29 | \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 83 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46912680491726189403346265781, −4.43959128520167464188661590950, −4.24642297469687367469854570734, −4.07085834810941884673309395035, −4.06290126220015714186479761468, −4.01628998277117597955479134618, −3.88833038792028134407616897199, −3.86561421433811943132965035200, −3.76021634913885828515263608812, −3.55245831630499198999208734966, −3.20307752340291635623029196937, −3.17325006952746617781311779603, −3.16359254788662527190539733778, −3.07466700820151543754262244854, −3.05764868649847414893347691191, −2.57856052430192880723067752895, −2.30666435097973230087939109912, −2.07942221783659488373078447965, −2.01518637056998855507874392963, −1.99189397916901956426965739668, −1.50269268871359637148224907544, −1.48160198417663858990829979556, −1.26407549998031595804509851284, −0.981773254448728305807019379431, −0.72410527832944070341814738241,
0.72410527832944070341814738241, 0.981773254448728305807019379431, 1.26407549998031595804509851284, 1.48160198417663858990829979556, 1.50269268871359637148224907544, 1.99189397916901956426965739668, 2.01518637056998855507874392963, 2.07942221783659488373078447965, 2.30666435097973230087939109912, 2.57856052430192880723067752895, 3.05764868649847414893347691191, 3.07466700820151543754262244854, 3.16359254788662527190539733778, 3.17325006952746617781311779603, 3.20307752340291635623029196937, 3.55245831630499198999208734966, 3.76021634913885828515263608812, 3.86561421433811943132965035200, 3.88833038792028134407616897199, 4.01628998277117597955479134618, 4.06290126220015714186479761468, 4.07085834810941884673309395035, 4.24642297469687367469854570734, 4.43959128520167464188661590950, 4.46912680491726189403346265781
Plot not available for L-functions of degree greater than 10.
