| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·15-s − 8·23-s + 3·25-s + 2·27-s − 2·31-s − 2·37-s − 6·45-s − 2·59-s + 8·67-s − 16·69-s + 2·71-s + 6·75-s + 81-s − 8·89-s − 4·93-s + 2·97-s − 4·111-s − 2·113-s + 16·115-s − 2·125-s + 127-s + 131-s − 4·135-s + 137-s + ⋯ |
| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·15-s − 8·23-s + 3·25-s + 2·27-s − 2·31-s − 2·37-s − 6·45-s − 2·59-s + 8·67-s − 16·69-s + 2·71-s + 6·75-s + 81-s − 8·89-s − 4·93-s + 2·97-s − 4·111-s − 2·113-s + 16·115-s − 2·125-s + 127-s + 131-s − 4·135-s + 137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3586720961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3586720961\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 5 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 7 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( ( 1 + T + T^{2} )^{8} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 37 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 59 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{8} \) |
| 71 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 83 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 89 | \( ( 1 + T + T^{2} )^{8} \) |
| 97 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
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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
−4.32801509692024875467776243308, −4.21668534274142585456052731911, −4.13712600598296971717671369452, −4.13077820116991630063198439689, −3.84433360353964124388669971458, −3.80840099383446542553098060016, −3.78567196601666207751237357000, −3.61729960179301366850380031045, −3.58418886820362785195632855123, −3.57700461717680160225671419087, −3.56303110613607760298212309982, −3.06333174787931476590933791131, −2.79575549184256168646708456511, −2.77923755223142338495361035367, −2.59608908758388173208369220750, −2.57990533760632707977610532407, −2.21110414080452641908912603621, −2.07539870427375014707055299856, −1.95909544567123710920095841052, −1.80045032489689858169181402559, −1.79232013696017646061738422107, −1.72894665450767901338040088902, −1.26294153761128970126630603489, −1.04052006312900831254152696419, −0.34730094615671320267926251175,
0.34730094615671320267926251175, 1.04052006312900831254152696419, 1.26294153761128970126630603489, 1.72894665450767901338040088902, 1.79232013696017646061738422107, 1.80045032489689858169181402559, 1.95909544567123710920095841052, 2.07539870427375014707055299856, 2.21110414080452641908912603621, 2.57990533760632707977610532407, 2.59608908758388173208369220750, 2.77923755223142338495361035367, 2.79575549184256168646708456511, 3.06333174787931476590933791131, 3.56303110613607760298212309982, 3.57700461717680160225671419087, 3.58418886820362785195632855123, 3.61729960179301366850380031045, 3.78567196601666207751237357000, 3.80840099383446542553098060016, 3.84433360353964124388669971458, 4.13077820116991630063198439689, 4.13712600598296971717671369452, 4.21668534274142585456052731911, 4.32801509692024875467776243308
Plot not available for L-functions of degree greater than 10.
