L(s) = 1 | − 5·13-s − 10·17-s − 32-s − 5·61-s − 5·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·221-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 5·13-s − 10·17-s − 32-s − 5·61-s − 5·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·221-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 101^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 101^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.007411272361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.007411272361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \) |
| 101 | \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \) |
good | 3 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 5 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 7 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 11 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 17 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{10} \) |
| 19 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 23 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 29 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 31 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 37 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 41 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 43 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 47 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 53 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 59 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 61 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 67 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 71 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 73 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 79 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 83 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 89 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 97 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.90988851759274814253596789412, −2.89064656875672964790906512934, −2.86116720594174462989549304218, −2.85356446163337558246027193514, −2.83036092763113532531951747623, −2.57495157628526989421961033027, −2.50593712585465730763233365284, −2.49576976475677993303283720544, −2.35699536337165054344293693292, −2.35680192243366114305958496144, −2.28410289493462730775820183978, −2.12865788153770767008274315388, −2.12855828021913603191146722350, −2.08708377605093776592735284629, −2.07572310612402283037005324535, −2.04627666451026924217809994555, −1.83708548387501276601184762813, −1.78528792672795120945510742226, −1.62026767266185514334382304882, −1.53361378080098745944304798823, −1.51516758300246311839125988170, −1.39312513326913642483377862286, −1.22195378149233920644209065716, −1.00930115635347560036163001700, −0.51712563085711301580363819109,
0.51712563085711301580363819109, 1.00930115635347560036163001700, 1.22195378149233920644209065716, 1.39312513326913642483377862286, 1.51516758300246311839125988170, 1.53361378080098745944304798823, 1.62026767266185514334382304882, 1.78528792672795120945510742226, 1.83708548387501276601184762813, 2.04627666451026924217809994555, 2.07572310612402283037005324535, 2.08708377605093776592735284629, 2.12855828021913603191146722350, 2.12865788153770767008274315388, 2.28410289493462730775820183978, 2.35680192243366114305958496144, 2.35699536337165054344293693292, 2.49576976475677993303283720544, 2.50593712585465730763233365284, 2.57495157628526989421961033027, 2.83036092763113532531951747623, 2.85356446163337558246027193514, 2.86116720594174462989549304218, 2.89064656875672964790906512934, 2.90988851759274814253596789412
Plot not available for L-functions of degree greater than 10.