| L(s) = 1 | + 4·2-s + 8·4-s + 12·8-s + 4·9-s + 15·16-s + 16·18-s − 16·29-s + 16·32-s + 32·36-s + 32·37-s + 32·53-s − 64·58-s − 16·61-s + 16·64-s + 48·72-s + 32·73-s + 128·74-s − 10·81-s − 16·89-s + 8·97-s + 128·106-s + 32·109-s + 32·113-s − 128·116-s − 64·122-s + 127-s + 8·128-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 4.24·8-s + 4/3·9-s + 15/4·16-s + 3.77·18-s − 2.97·29-s + 2.82·32-s + 16/3·36-s + 5.26·37-s + 4.39·53-s − 8.40·58-s − 2.04·61-s + 2·64-s + 5.65·72-s + 3.74·73-s + 14.8·74-s − 1.11·81-s − 1.69·89-s + 0.812·97-s + 12.4·106-s + 3.06·109-s + 3.01·113-s − 11.8·116-s − 5.79·122-s + 0.0887·127-s + 0.707·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(50.45209926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(50.45209926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + p^{5} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 - 2 T^{2} + 11 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 31 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 18 T^{4} + 2291 T^{8} - 18 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( 1 - 100 T^{4} - 986 T^{8} - 100 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 p T^{2} + 579 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 220 T^{4} + 67942 T^{8} - 220 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 72 T^{2} + 2322 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 60 T^{4} + 944774 T^{8} - 60 p^{4} T^{12} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 960 T^{3} + 6671 T^{4} - 960 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 114 T^{2} + 6939 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 7022 T^{4} + 21833011 T^{8} + 7022 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 8 T + 104 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 2140 T^{4} - 6110426 T^{8} - 2140 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 12860 T^{4} + 79679014 T^{8} + 12860 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( 1 - 12210 T^{4} + 75514259 T^{8} - 12210 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 656 T^{3} + 2338 T^{4} - 656 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 224 T^{2} + 22978 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 5340 T^{4} - 15919258 T^{8} + 5340 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 8 T + 32 T^{2} + 488 T^{3} + 6658 T^{4} + 488 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4 T + 8 T^{2} - 380 T^{3} + 18046 T^{4} - 380 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.69332039616743638885945727302, −4.32678321895693776320916030225, −4.24484548864085484044290783875, −4.13013918729079653301970492440, −4.05187989332398139833994154622, −4.04154618687012710832250013428, −3.81433063941378150952798439949, −3.69206611723015417721006910360, −3.60348508708329909913159008595, −3.50740914284846483776822259190, −3.12926353832668003628520837604, −3.09685256781746690005287938480, −2.84384355152831518066064766462, −2.62721881701949230500084799375, −2.62660857587635848934185378297, −2.61208287021843563986980458385, −2.12634056120110779184721200649, −2.07167449740982803780460512215, −1.92126103466248335712088062314, −1.73349338169164008429971313769, −1.64930039668157259843887270811, −1.21255122120626676098083056232, −0.904232618443666208740626391944, −0.64916909191317858139326362485, −0.63913290062615056275473247827,
0.63913290062615056275473247827, 0.64916909191317858139326362485, 0.904232618443666208740626391944, 1.21255122120626676098083056232, 1.64930039668157259843887270811, 1.73349338169164008429971313769, 1.92126103466248335712088062314, 2.07167449740982803780460512215, 2.12634056120110779184721200649, 2.61208287021843563986980458385, 2.62660857587635848934185378297, 2.62721881701949230500084799375, 2.84384355152831518066064766462, 3.09685256781746690005287938480, 3.12926353832668003628520837604, 3.50740914284846483776822259190, 3.60348508708329909913159008595, 3.69206611723015417721006910360, 3.81433063941378150952798439949, 4.04154618687012710832250013428, 4.05187989332398139833994154622, 4.13013918729079653301970492440, 4.24484548864085484044290783875, 4.32678321895693776320916030225, 4.69332039616743638885945727302
Plot not available for L-functions of degree greater than 10.
