| L(s) = 1 | + 8·3-s + 36·9-s + 8·11-s + 16·25-s + 120·27-s + 64·33-s − 16·49-s + 32·59-s − 64·67-s + 128·75-s + 330·81-s − 32·89-s + 64·97-s + 288·99-s − 32·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 128·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 12·9-s + 2.41·11-s + 16/5·25-s + 23.0·27-s + 11.1·33-s − 2.28·49-s + 4.16·59-s − 7.81·67-s + 14.7·75-s + 36.6·81-s − 3.39·89-s + 6.49·97-s + 28.9·99-s − 3.01·113-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-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
| \(L(1)\) |
\(\approx\) |
\(25.09802411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.09802411\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{8} \) |
| 11 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 8 T^{2} + 94 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 574 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 12 T^{2} + 294 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 36 T^{2} + 726 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 76 T^{2} + 2806 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 84 T^{2} + 4806 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 132 T^{2} + 7734 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 160 T^{2} + 10638 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 61 | \( ( 1 + 144 T^{2} + 10206 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 96 T^{2} + 3566 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 212 T^{2} + 21574 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 24 T^{2} + 12606 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 132 T^{2} + 17814 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.72957094018226333654987165615, −3.48021769689897485918527421640, −3.44819992790054438287451694819, −3.41743218155382442642578894314, −3.30760033528959452075878587275, −3.23502883704045642468945647086, −3.13397924312298050523042460835, −2.99801157328739761078437349148, −2.82305514478702160125491784618, −2.70999753358647604582109294473, −2.64162076381377477662776735215, −2.52454661162660402551971693495, −2.24077778079260068514273026912, −2.14574970346148050745634299596, −2.07303504847157466126494277564, −1.97756425585687873960851387364, −1.84405539217763664348262954973, −1.50487140620469016976157397499, −1.42963995576102604892506955568, −1.35303620115156347109209667354, −1.19333983112247701079599101677, −1.06846437398289278528684945330, −0.913211565520439327764649421529, −0.70432919958612635822759674867, −0.11603836261291332019293115594,
0.11603836261291332019293115594, 0.70432919958612635822759674867, 0.913211565520439327764649421529, 1.06846437398289278528684945330, 1.19333983112247701079599101677, 1.35303620115156347109209667354, 1.42963995576102604892506955568, 1.50487140620469016976157397499, 1.84405539217763664348262954973, 1.97756425585687873960851387364, 2.07303504847157466126494277564, 2.14574970346148050745634299596, 2.24077778079260068514273026912, 2.52454661162660402551971693495, 2.64162076381377477662776735215, 2.70999753358647604582109294473, 2.82305514478702160125491784618, 2.99801157328739761078437349148, 3.13397924312298050523042460835, 3.23502883704045642468945647086, 3.30760033528959452075878587275, 3.41743218155382442642578894314, 3.44819992790054438287451694819, 3.48021769689897485918527421640, 3.72957094018226333654987165615
Plot not available for L-functions of degree greater than 10.
