| L(s) = 1 | − 8·13-s + 8·37-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 + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 8·13-s + 8·37-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 + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6391169577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6391169577\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T^{8} \) |
| good | 3 | \( 1 + T^{16} \) |
| 5 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{2} )^{8} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.72927509088552573284648618315, −3.67257101686677322232805725383, −3.47335404201374123687714644612, −3.35843401669977870108536445147, −3.19572815556730129968096934807, −3.19463920623852222311775881511, −3.05561388486647462761963313863, −2.78429800602625943931861210239, −2.71885843686669212073336595609, −2.65517079626516605777215294210, −2.64194241934139287759876031297, −2.53515581371718937145266410013, −2.30545358562821348702677656642, −2.24625084026719771203152254525, −2.09187156278542548737629572141, −2.09009878459122569250476105671, −2.05701410331284451040445214289, −2.03036142762918088964082472660, −1.43344633053361596346928323321, −1.19211359868580454245094185473, −1.17028550969851804372648981741, −1.12894925946816484229440663141, −0.805824639668741540567218698872, −0.41489849761726085974518502606, −0.36059908334400582422416166607,
0.36059908334400582422416166607, 0.41489849761726085974518502606, 0.805824639668741540567218698872, 1.12894925946816484229440663141, 1.17028550969851804372648981741, 1.19211359868580454245094185473, 1.43344633053361596346928323321, 2.03036142762918088964082472660, 2.05701410331284451040445214289, 2.09009878459122569250476105671, 2.09187156278542548737629572141, 2.24625084026719771203152254525, 2.30545358562821348702677656642, 2.53515581371718937145266410013, 2.64194241934139287759876031297, 2.65517079626516605777215294210, 2.71885843686669212073336595609, 2.78429800602625943931861210239, 3.05561388486647462761963313863, 3.19463920623852222311775881511, 3.19572815556730129968096934807, 3.35843401669977870108536445147, 3.47335404201374123687714644612, 3.67257101686677322232805725383, 3.72927509088552573284648618315
Plot not available for L-functions of degree greater than 10.
