| L(s) = 1 | − 12·19-s + 3·25-s − 4·31-s + 4·49-s + 28·61-s − 48·79-s − 56·109-s − 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.75·19-s + 3/5·25-s − 0.718·31-s + 4/7·49-s + 3.58·61-s − 5.40·79-s − 5.36·109-s − 3.81·121-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 + 2/13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05682076014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05682076014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 3 T^{2} + 4 p T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 - 2 T^{2} - 30 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T^{2} + 48 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 15 T^{2} + 602 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 24 T^{2} + 686 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 59 T^{2} + 3318 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 129 T^{2} + 7232 T^{4} + 129 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 146 T^{2} + 8898 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 18 T^{2} + 1274 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 134 T^{2} + 8946 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 3 p T^{2} + 13988 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7 T + 102 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8 T^{2} - 3906 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 53 T^{2} + 3528 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 227 T^{2} + 22734 T^{4} - 227 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 210 T^{2} + 23642 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 246 T^{2} + 28907 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 182 T^{2} + 16650 T^{4} - 182 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.97279057206456251125364689685, −3.92270048991999864574954141683, −3.80641421395681965773113392101, −3.74373155189888670241321788738, −3.61056285882823967062008888626, −3.14467889491210168575978120433, −3.10708088473412889446418952197, −3.07007998573965699000917187943, −2.98677255989246034600843760227, −2.80255466588719061263647788646, −2.74231423907335546240935448168, −2.48124376878164491183308734543, −2.31357399947324902391854163187, −2.27501439770638048899578206733, −2.08462541596955301521802529456, −2.04120237246042569486673448670, −1.97695313580484505490615316462, −1.52450434883807959672339501111, −1.31281490460219658780685591310, −1.29008833056833967569872976249, −1.21365738161449770588939694300, −1.05421333218261711688018379341, −0.69088774826727342543938868381, −0.14230163989165245450413356432, −0.06794938029132629590035500001,
0.06794938029132629590035500001, 0.14230163989165245450413356432, 0.69088774826727342543938868381, 1.05421333218261711688018379341, 1.21365738161449770588939694300, 1.29008833056833967569872976249, 1.31281490460219658780685591310, 1.52450434883807959672339501111, 1.97695313580484505490615316462, 2.04120237246042569486673448670, 2.08462541596955301521802529456, 2.27501439770638048899578206733, 2.31357399947324902391854163187, 2.48124376878164491183308734543, 2.74231423907335546240935448168, 2.80255466588719061263647788646, 2.98677255989246034600843760227, 3.07007998573965699000917187943, 3.10708088473412889446418952197, 3.14467889491210168575978120433, 3.61056285882823967062008888626, 3.74373155189888670241321788738, 3.80641421395681965773113392101, 3.92270048991999864574954141683, 3.97279057206456251125364689685
Plot not available for L-functions of degree greater than 10.
