| L(s) = 1 | − 3·4-s + 6·16-s − 6·19-s − 10·64-s + 18·76-s + 6·121-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 + ⋯ |
| L(s) = 1 | − 3·4-s + 6·16-s − 6·19-s − 10·64-s + 18·76-s + 6·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4340160515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4340160515\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - T^{6} + T^{12} \) |
| 7 | \( 1 - T^{6} + T^{12} \) |
| 11 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 13 | \( 1 - T^{6} + T^{12} \) |
| 17 | \( 1 - T^{6} + T^{12} \) |
| 23 | \( 1 - T^{6} + T^{12} \) |
| 29 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 41 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 43 | \( ( 1 + T^{2} )^{6} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 53 | \( 1 - T^{6} + T^{12} \) |
| 59 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 67 | \( 1 - T^{6} + T^{12} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( 1 - T^{6} + T^{12} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( ( 1 + T^{2} )^{6} \) |
| 89 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 97 | \( ( 1 + T^{2} )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.71758776344464674205871723641, −4.30327564757625793169181641224, −4.23774687609070138180374282642, −4.11195832858536234956969784778, −4.08343082045360354246643870477, −4.01147978866766937997893607457, −3.99365414189022273243799070410, −3.53530464658958635503126555797, −3.50683843871902413610775112285, −3.43340084325485757930990958980, −3.22465770066772047932746935620, −3.02063651519615724527946498287, −2.78718196518846251748011518083, −2.76541946939768003642827771560, −2.35101914896618355435565505983, −2.28381096615094497585937348680, −2.21520706055977960957688491061, −1.95578864965617538534759869238, −1.70478601775394990787670295736, −1.68525463392386657561590875773, −1.31404186641320955428549139283, −1.29774912106516932850348464564, −0.59391308421665691461222734084, −0.56370632011863495275363234968, −0.42637819116909785239484517734,
0.42637819116909785239484517734, 0.56370632011863495275363234968, 0.59391308421665691461222734084, 1.29774912106516932850348464564, 1.31404186641320955428549139283, 1.68525463392386657561590875773, 1.70478601775394990787670295736, 1.95578864965617538534759869238, 2.21520706055977960957688491061, 2.28381096615094497585937348680, 2.35101914896618355435565505983, 2.76541946939768003642827771560, 2.78718196518846251748011518083, 3.02063651519615724527946498287, 3.22465770066772047932746935620, 3.43340084325485757930990958980, 3.50683843871902413610775112285, 3.53530464658958635503126555797, 3.99365414189022273243799070410, 4.01147978866766937997893607457, 4.08343082045360354246643870477, 4.11195832858536234956969784778, 4.23774687609070138180374282642, 4.30327564757625793169181641224, 4.71758776344464674205871723641
Plot not available for L-functions of degree greater than 10.
