| L(s) = 1 | − 8·43-s − 8·49-s − 2·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 | − 8·43-s − 8·49-s − 2·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2727602442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2727602442\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 13 | \( ( 1 + T^{4} )^{2} \) |
| good | 5 | \( ( 1 + T^{8} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{8} \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 19 | \( ( 1 + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{2} )^{8} \) |
| 29 | \( ( 1 + T^{4} )^{4} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 47 | \( ( 1 + T^{8} )^{2} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 + T^{2} )^{8} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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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
−4.95453201455515120322607667958, −4.83691777022271644997221636205, −4.69170158630456900308473115441, −4.54734357132608278599234076129, −4.27922511120601303739164792078, −4.27391684529640215235864146333, −4.15941798208239544213654513720, −3.96959940041395700049470072592, −3.49626980091899211900167495765, −3.47568573785165170670278817830, −3.42137822706251170721136390781, −3.35600686624839614393160237109, −3.32708304348950427058141625128, −3.18891818677673892125455825707, −2.85007558694573951370941669280, −2.74691665918881862604545528709, −2.56558107265464721939571014085, −2.53530285833621762441390648403, −1.87201397448150074188789799761, −1.74446295348748728551273560509, −1.70273815481864581087862194144, −1.68807888608735225006993147654, −1.61868251366773896733852901583, −1.39352023955886229394312368819, −0.56799868889114961379323554390,
0.56799868889114961379323554390, 1.39352023955886229394312368819, 1.61868251366773896733852901583, 1.68807888608735225006993147654, 1.70273815481864581087862194144, 1.74446295348748728551273560509, 1.87201397448150074188789799761, 2.53530285833621762441390648403, 2.56558107265464721939571014085, 2.74691665918881862604545528709, 2.85007558694573951370941669280, 3.18891818677673892125455825707, 3.32708304348950427058141625128, 3.35600686624839614393160237109, 3.42137822706251170721136390781, 3.47568573785165170670278817830, 3.49626980091899211900167495765, 3.96959940041395700049470072592, 4.15941798208239544213654513720, 4.27391684529640215235864146333, 4.27922511120601303739164792078, 4.54734357132608278599234076129, 4.69170158630456900308473115441, 4.83691777022271644997221636205, 4.95453201455515120322607667958
Plot not available for L-functions of degree greater than 10.
