| L(s) = 1 | + 2·3-s + 4·7-s + 4·9-s + 8·13-s + 16·19-s + 8·21-s + 4·27-s − 8·31-s − 28·37-s + 16·39-s − 36·43-s + 8·49-s + 32·57-s + 28·61-s + 16·63-s + 40·67-s − 28·73-s − 16·79-s + 5·81-s + 32·91-s − 16·93-s + 20·97-s − 64·109-s − 56·111-s + 32·117-s − 24·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 4/3·9-s + 2.21·13-s + 3.67·19-s + 1.74·21-s + 0.769·27-s − 1.43·31-s − 4.60·37-s + 2.56·39-s − 5.48·43-s + 8/7·49-s + 4.23·57-s + 3.58·61-s + 2.01·63-s + 4.88·67-s − 3.27·73-s − 1.80·79-s + 5/9·81-s + 3.35·91-s − 1.65·93-s + 2.03·97-s − 6.13·109-s − 5.31·111-s + 2.95·117-s − 2.18·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.945176367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.945176367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( 1 - 22 T^{4} + 939 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} + 24 T^{3} - 73 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 24 T^{2} + 338 T^{4} + 3504 T^{6} + 29907 T^{8} + 3504 p^{2} T^{10} + 338 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 38 T^{2} + 613 T^{4} - 9614 T^{6} + 189724 T^{8} - 9614 p^{2} T^{10} + 613 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 8 T + 20 T^{2} + 60 T^{3} - 649 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 34 T^{2} - 707 T^{4} + 6154 T^{6} + 1791292 T^{8} + 6154 p^{2} T^{10} - 707 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 8 T^{2} + 36 T^{3} - 322 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 14 T + 113 T^{2} + 18 p T^{3} + 104 p T^{4} + 18 p^{2} T^{5} + 113 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 54 T^{2} + 221 T^{4} + 40554 T^{6} - 627828 T^{8} + 40554 p^{2} T^{10} + 221 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 18 T + 212 T^{2} + 1872 T^{3} + 13611 T^{4} + 1872 p T^{5} + 212 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 190 T^{2} + 14535 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 24 T^{2} - 3202 T^{4} - 81456 T^{6} + 70227 T^{8} - 81456 p^{2} T^{10} - 3202 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 20 T + 164 T^{2} - 564 T^{3} + 359 T^{4} - 564 p T^{5} + 164 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 24 T^{2} - 5554 T^{4} + 137904 T^{6} + 8572707 T^{8} + 137904 p^{2} T^{10} - 5554 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 24 T^{2} + 9026 T^{4} - 212016 T^{6} + 16818147 T^{8} - 212016 p^{2} T^{10} + 9026 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 10 T + 2 p T^{2} - 2124 T^{3} + 23279 T^{4} - 2124 p T^{5} + 2 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.79988950026954606932974704777, −4.65045796969550341235098523409, −4.11368456079074009281044834302, −3.99638463609168009833063419066, −3.89607773803268357450881601478, −3.87435732304216713120584090101, −3.72044108469972944741742403507, −3.66033525275092640227061942354, −3.60481636612753592979344563431, −3.53118871726431835882892173921, −3.22018295799705620681669673962, −3.13402229082820806357964382122, −2.83573064122903657920032121676, −2.71955314992746300220660379646, −2.52906036445032653005205020099, −2.52865162578355375189866279071, −2.17633209303968847251292465081, −1.81707856200182267366479567965, −1.57482619999457212044462351694, −1.56227927009151504941502874209, −1.55481508596812959132166508044, −1.49650738571084111305570000705, −1.00284354728127132521567759653, −0.990322527641059224456056828872, −0.16621611118622402308105114894,
0.16621611118622402308105114894, 0.990322527641059224456056828872, 1.00284354728127132521567759653, 1.49650738571084111305570000705, 1.55481508596812959132166508044, 1.56227927009151504941502874209, 1.57482619999457212044462351694, 1.81707856200182267366479567965, 2.17633209303968847251292465081, 2.52865162578355375189866279071, 2.52906036445032653005205020099, 2.71955314992746300220660379646, 2.83573064122903657920032121676, 3.13402229082820806357964382122, 3.22018295799705620681669673962, 3.53118871726431835882892173921, 3.60481636612753592979344563431, 3.66033525275092640227061942354, 3.72044108469972944741742403507, 3.87435732304216713120584090101, 3.89607773803268357450881601478, 3.99638463609168009833063419066, 4.11368456079074009281044834302, 4.65045796969550341235098523409, 4.79988950026954606932974704777
Plot not available for L-functions of degree greater than 10.
