| L(s) = 1 | + 100·25-s − 110·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 130·169-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 | + 20·25-s − 10·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 10·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(105.9146528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(105.9146528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 53 T^{10} + p^{10} T^{20} \) |
| 11 | \( ( 1 + p T^{2} )^{10} \) |
| 13 | \( ( 1 - p T^{2} )^{10} \) |
| good | 3 | \( ( 1 - 20 T^{5} + p^{5} T^{10} )^{2}( 1 + 20 T^{5} + p^{5} T^{10} )^{2} \) |
| 5 | \( ( 1 - p T^{2} )^{20} \) |
| 7 | \( ( 1 - 1538 T^{10} + p^{10} T^{20} )^{2} \) |
| 17 | \( ( 1 - p T^{2} )^{20} \) |
| 19 | \( ( 1 - 4932794 T^{10} + p^{10} T^{20} )^{2} \) |
| 23 | \( ( 1 - 4728 T^{5} + p^{5} T^{10} )^{2}( 1 + 4728 T^{5} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 - p T^{2} )^{20} \) |
| 31 | \( ( 1 + p T^{2} )^{20} \) |
| 37 | \( ( 1 - p T^{2} )^{20} \) |
| 41 | \( ( 1 - 231619298 T^{10} + p^{10} T^{20} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{20} \) |
| 47 | \( ( 1 + p T^{2} )^{20} \) |
| 53 | \( ( 1 + 24858 T^{5} + p^{5} T^{10} )^{4} \) |
| 59 | \( ( 1 + p T^{2} )^{20} \) |
| 61 | \( ( 1 - p T^{2} )^{20} \) |
| 67 | \( ( 1 + p T^{2} )^{20} \) |
| 71 | \( ( 1 + p T^{2} )^{20} \) |
| 73 | \( ( 1 - 3615121442 T^{10} + p^{10} T^{20} )^{2} \) |
| 79 | \( ( 1 + p T^{2} )^{20} \) |
| 83 | \( ( 1 - 7348587002 T^{10} + p^{10} T^{20} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{20} \) |
| 97 | \( ( 1 - p T^{2} )^{20} \) |
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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
−2.54969528765480435911599659254, −2.46729648251359805965006727976, −2.45774027981099430087042822004, −2.38718570401272644688637917659, −2.28091824620349044611632945713, −2.24153799509940681444339143928, −2.13937067100749525007343031336, −1.91703279436532793319421368390, −1.87590730601787830764774378366, −1.62234578428158458820098090406, −1.55989435555286357883205314812, −1.52872909918011123835313122800, −1.45653785920000254702556262497, −1.41458964978853597042199183405, −1.39442563929003610132524544808, −1.37245975661169595968541938677, −1.22077482930444360939177333025, −0.943373645395302133834869649644, −0.921942173633447648974562236669, −0.809833474543868119328640499451, −0.76597355273803042660232971525, −0.76195282863172093892372058786, −0.73191051963830214474175738247, −0.49320701665392058677126602891, −0.35696472529627918496203528366,
0.35696472529627918496203528366, 0.49320701665392058677126602891, 0.73191051963830214474175738247, 0.76195282863172093892372058786, 0.76597355273803042660232971525, 0.809833474543868119328640499451, 0.921942173633447648974562236669, 0.943373645395302133834869649644, 1.22077482930444360939177333025, 1.37245975661169595968541938677, 1.39442563929003610132524544808, 1.41458964978853597042199183405, 1.45653785920000254702556262497, 1.52872909918011123835313122800, 1.55989435555286357883205314812, 1.62234578428158458820098090406, 1.87590730601787830764774378366, 1.91703279436532793319421368390, 2.13937067100749525007343031336, 2.24153799509940681444339143928, 2.28091824620349044611632945713, 2.38718570401272644688637917659, 2.45774027981099430087042822004, 2.46729648251359805965006727976, 2.54969528765480435911599659254
Plot not available for L-functions of degree greater than 10.
