| L(s) = 1 | − 8·4-s + 36·16-s + 8·49-s − 120·64-s + 32·79-s + 20·81-s − 32·109-s + 112·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 160·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 64·196-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4·4-s + 9·16-s + 8/7·49-s − 15·64-s + 3.60·79-s + 20/9·81-s − 3.06·109-s + 10.1·121-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 + 12.3·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4.57·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1243907469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1243907469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( ( 1 - 10 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 4 T^{2} - 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| good | 11 | \( ( 1 - 6 T + p T^{2} )^{8}( 1 + 6 T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 40 T^{2} + 710 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 + 44 T^{2} + 950 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 24 T^{2} + 838 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 28 T^{2} + 806 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 52 T^{2} + 1350 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 84 T^{2} + 3574 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + 20 T^{2} + 38 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 84 T^{2} + 4678 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 44 T^{2} + 2390 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 108 T^{2} + 6886 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 20 T^{2} + 3926 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 216 T^{2} + 18598 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 136 T^{2} + 9798 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 140 T^{2} + 12086 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 28 T^{2} + 9270 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 252 T^{2} + 26422 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 + 224 T^{2} + 26294 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 60 T^{2} + 14950 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 108 T^{2} + 16246 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.53903713272494110986143488104, −2.51792064107248490324160511821, −2.43021840467089211455405742856, −2.35881035016493193817386677824, −2.30844436196626775003000930536, −2.12873050102150763882118144543, −2.02361111086540516293672317346, −1.93034533380014404218023767797, −1.91410671156693025954759156710, −1.90912259015507008450464035621, −1.70516501590312646831251703049, −1.63594264162412229358345708799, −1.59498779195073362440153173897, −1.48159261463078389347192203210, −1.23389109159400260357459850646, −1.13574900802858243086407371277, −1.01286610416787237095324813392, −0.989153278428979362414854182035, −0.924246234439513960075637866288, −0.78060940840562195400448870288, −0.74156371520969303179402116821, −0.61244916320849836865222375456, −0.42768127843963389933949620909, −0.12073940989756411574864509315, −0.079194181700877280792550029491,
0.079194181700877280792550029491, 0.12073940989756411574864509315, 0.42768127843963389933949620909, 0.61244916320849836865222375456, 0.74156371520969303179402116821, 0.78060940840562195400448870288, 0.924246234439513960075637866288, 0.989153278428979362414854182035, 1.01286610416787237095324813392, 1.13574900802858243086407371277, 1.23389109159400260357459850646, 1.48159261463078389347192203210, 1.59498779195073362440153173897, 1.63594264162412229358345708799, 1.70516501590312646831251703049, 1.90912259015507008450464035621, 1.91410671156693025954759156710, 1.93034533380014404218023767797, 2.02361111086540516293672317346, 2.12873050102150763882118144543, 2.30844436196626775003000930536, 2.35881035016493193817386677824, 2.43021840467089211455405742856, 2.51792064107248490324160511821, 2.53903713272494110986143488104
Plot not available for L-functions of degree greater than 10.
