| L(s) = 1 | − 8·29-s − 8·53-s + 8·73-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 | − 8·29-s − 8·53-s + 8·73-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9681592396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9681592396\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T^{8} \) |
| 17 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( 1 + T^{16} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T )^{8}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 - T )^{8}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−3.98481383930466882922556282336, −3.44510155467515789533213664601, −3.41146852683877591599247077537, −3.38818298906312690017458328124, −3.37963004911583003169785342440, −3.34149881566140933994300280591, −3.28560208209169558538200393017, −3.24866189831341189291759586131, −3.14992827310433746175800624766, −2.59813650817866102043056173567, −2.51976408056421402841625587788, −2.43286156854930447237044888127, −2.37688967748981707911616981770, −2.23212931891974965047067550096, −2.19653195236309198564620768210, −1.86250414714494989085157435507, −1.73304213756173199190955026389, −1.65936968652517644180336779397, −1.62782347665423804018228569768, −1.57835454000680345116078024617, −1.50068695231121798881048538261, −1.11664830255080215145660460918, −0.63586919680076309071024418276, −0.59182908588596048626385404265, −0.37332022324572952706584745994,
0.37332022324572952706584745994, 0.59182908588596048626385404265, 0.63586919680076309071024418276, 1.11664830255080215145660460918, 1.50068695231121798881048538261, 1.57835454000680345116078024617, 1.62782347665423804018228569768, 1.65936968652517644180336779397, 1.73304213756173199190955026389, 1.86250414714494989085157435507, 2.19653195236309198564620768210, 2.23212931891974965047067550096, 2.37688967748981707911616981770, 2.43286156854930447237044888127, 2.51976408056421402841625587788, 2.59813650817866102043056173567, 3.14992827310433746175800624766, 3.24866189831341189291759586131, 3.28560208209169558538200393017, 3.34149881566140933994300280591, 3.37963004911583003169785342440, 3.38818298906312690017458328124, 3.41146852683877591599247077537, 3.44510155467515789533213664601, 3.98481383930466882922556282336
Plot not available for L-functions of degree greater than 10.
