| L(s) = 1 | + 96·19-s + 56·25-s + 160·43-s + 152·49-s + 192·67-s + 304·73-s + 112·97-s − 520·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 312·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 5.05·19-s + 2.23·25-s + 3.72·43-s + 3.10·49-s + 2.86·67-s + 4.16·73-s + 1.15·97-s − 4.29·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.84·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(74.91962893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(74.91962893\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 28 T^{2} + 22 p^{2} T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 76 T^{2} + 5350 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 12 p T^{2} + 30950 T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 868 T^{2} + 341062 T^{4} + 868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 24 T + 642 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 196 T^{2} - 348218 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2524 T^{2} + 2963302 T^{4} - 2524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3084 T^{2} + 4152230 T^{4} - 3084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 5020 T^{2} + 10044838 T^{4} - 5020 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 3364 T^{2} + 7778182 T^{4} + 3364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 40 T + 3874 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 252 T^{2} + 1517702 T^{4} + 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4060 T^{2} + 11815462 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 8932 T^{2} + 40509862 T^{4} + 8932 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14044 T^{2} + 76824550 T^{4} - 14044 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 48 T + 1490 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 76 T + 4038 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 17228 T^{2} + 145319334 T^{4} - 17228 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 10948 T^{2} + 121211302 T^{4} + 10948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 11076 T^{2} + 54999110 T^{4} + 11076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 28 T + 15430 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.46730469590373822378931362385, −3.36161850196035834893992217441, −3.26674404620747507996881138663, −3.16605739091386625737379380351, −2.95790931610458378514375765480, −2.92127205934222467628356453121, −2.89866153606556323683837545820, −2.85695673754211737705326189626, −2.72877056198714328871784590953, −2.34769993846576267440093229635, −2.27272910413048693885637413681, −2.16874663811597812984495556167, −2.16008208256046431292511224820, −1.91326545978170451085465371781, −1.68539117870889656938406947335, −1.57869897821058179077864820931, −1.57684433859005953747986112550, −1.04903794999320851483784304221, −0.933309086868771325246903035922, −0.915520291802682587465827478807, −0.816432090833609849843571168446, −0.77717641527723026535037417152, −0.75160039240003863071761584682, −0.45966142290642423231133171692, −0.28742234718302512122044813532,
0.28742234718302512122044813532, 0.45966142290642423231133171692, 0.75160039240003863071761584682, 0.77717641527723026535037417152, 0.816432090833609849843571168446, 0.915520291802682587465827478807, 0.933309086868771325246903035922, 1.04903794999320851483784304221, 1.57684433859005953747986112550, 1.57869897821058179077864820931, 1.68539117870889656938406947335, 1.91326545978170451085465371781, 2.16008208256046431292511224820, 2.16874663811597812984495556167, 2.27272910413048693885637413681, 2.34769993846576267440093229635, 2.72877056198714328871784590953, 2.85695673754211737705326189626, 2.89866153606556323683837545820, 2.92127205934222467628356453121, 2.95790931610458378514375765480, 3.16605739091386625737379380351, 3.26674404620747507996881138663, 3.36161850196035834893992217441, 3.46730469590373822378931362385
Plot not available for L-functions of degree greater than 10.
