| L(s) = 1 | + 4·25-s + 16·49-s − 16·61-s − 16·109-s − 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4/5·25-s + 16/7·49-s − 2.04·61-s − 1.53·109-s − 5.81·121-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 + 6.15·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799908713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799908713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.59757925613144105902620780351, −4.22934588629934257059412812236, −4.18983299829703950659083867091, −4.11316086485883064950339852275, −4.01673308301056584107856853347, −3.97907840386382966471183928470, −3.69746539392307751751406557293, −3.49727725006992297713349159886, −3.39002847979506869504552085278, −3.27419692810514659180704626670, −3.06141871248123907717301273323, −2.96844952529009270716045107133, −2.88778834536485631000205573100, −2.51141642770941119066743926451, −2.44614347347559622644770360676, −2.28156256289423438091092751051, −2.27296326899149391535151945752, −2.13412343188451487634087167062, −1.60862282540554099598551133693, −1.41028965816569564585058153464, −1.35791019854649732726761464363, −1.31244655177094474016514722907, −0.943702108044158655222633330718, −0.48166225715866406438412163783, −0.29659754096010317512968312714,
0.29659754096010317512968312714, 0.48166225715866406438412163783, 0.943702108044158655222633330718, 1.31244655177094474016514722907, 1.35791019854649732726761464363, 1.41028965816569564585058153464, 1.60862282540554099598551133693, 2.13412343188451487634087167062, 2.27296326899149391535151945752, 2.28156256289423438091092751051, 2.44614347347559622644770360676, 2.51141642770941119066743926451, 2.88778834536485631000205573100, 2.96844952529009270716045107133, 3.06141871248123907717301273323, 3.27419692810514659180704626670, 3.39002847979506869504552085278, 3.49727725006992297713349159886, 3.69746539392307751751406557293, 3.97907840386382966471183928470, 4.01673308301056584107856853347, 4.11316086485883064950339852275, 4.18983299829703950659083867091, 4.22934588629934257059412812236, 4.59757925613144105902620780351
Plot not available for L-functions of degree greater than 10.
