| L(s) = 1 | + 4·9-s − 16·25-s − 16·49-s − 6·81-s − 128·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 64·225-s + 227-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 3.19·25-s − 2.28·49-s − 2/3·81-s − 12.9·97-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 4.26·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03047303413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03047303413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 16 T + p T^{2} )^{8} \) |
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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
−3.87965314282226652032055748291, −3.75736874156583897782976151672, −3.62049816304475346789696317276, −3.57688953087101023151301802739, −3.31262698730929541103031956008, −3.20214024079631095227704214307, −3.08907445754423512692829877858, −2.97151180654425763367627678317, −2.72647390429033586898653842753, −2.64529821567999333603398856985, −2.56530699336184026541890768729, −2.40609184890730256468136172492, −2.28255437598991488704209352615, −2.26784892294191879686946799166, −1.90067577625380079602012023587, −1.81036806428486112348093655570, −1.65191075414687049626815233700, −1.54177794209920300594389668452, −1.26668958981747295618189006950, −1.20892338950432415457636157290, −1.19942881195055468853910963359, −1.16425056984041805573984713890, −0.52578378829343161645694512547, −0.15410837460406877111786265394, −0.04021834314813098714074605150,
0.04021834314813098714074605150, 0.15410837460406877111786265394, 0.52578378829343161645694512547, 1.16425056984041805573984713890, 1.19942881195055468853910963359, 1.20892338950432415457636157290, 1.26668958981747295618189006950, 1.54177794209920300594389668452, 1.65191075414687049626815233700, 1.81036806428486112348093655570, 1.90067577625380079602012023587, 2.26784892294191879686946799166, 2.28255437598991488704209352615, 2.40609184890730256468136172492, 2.56530699336184026541890768729, 2.64529821567999333603398856985, 2.72647390429033586898653842753, 2.97151180654425763367627678317, 3.08907445754423512692829877858, 3.20214024079631095227704214307, 3.31262698730929541103031956008, 3.57688953087101023151301802739, 3.62049816304475346789696317276, 3.75736874156583897782976151672, 3.87965314282226652032055748291
Plot not available for L-functions of degree greater than 10.
