| L(s) = 1 | + 2·3-s + 4·5-s − 2·7-s + 9-s + 11-s + 10·13-s + 8·15-s − 3·19-s − 4·21-s + 2·23-s + 10·25-s − 2·27-s + 6·29-s − 14·31-s + 2·33-s − 8·35-s + 7·37-s + 20·39-s − 2·41-s − 7·43-s + 4·45-s + 2·47-s + 15·49-s + 22·53-s + 4·55-s − 6·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 2.77·13-s + 2.06·15-s − 0.688·19-s − 0.872·21-s + 0.417·23-s + 2·25-s − 0.384·27-s + 1.11·29-s − 2.51·31-s + 0.348·33-s − 1.35·35-s + 1.15·37-s + 3.20·39-s − 0.312·41-s − 1.06·43-s + 0.596·45-s + 0.291·47-s + 15/7·49-s + 3.02·53-s + 0.539·55-s − 0.794·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.40854656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.40854656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 7 T^{2} + 14 T^{3} - 68 T^{4} + 14 p T^{5} - 7 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3 T - 17 T^{2} - 36 T^{3} + 144 T^{4} - 36 p T^{5} - 17 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T + 14 T^{2} + 112 T^{3} - 521 T^{4} + 112 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 26 T^{2} + 288 T^{3} - 1785 T^{4} + 288 p T^{5} + 26 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 7 T - 23 T^{2} + 14 T^{3} + 2002 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T - 22 T^{2} - 112 T^{3} - 1169 T^{4} - 112 p T^{5} - 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 7 T - 35 T^{2} - 14 T^{3} + 3100 T^{4} - 14 p T^{5} - 35 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - T + 80 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 11 T + 122 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 86 T^{2} - 112 T^{3} - 2945 T^{4} - 112 p T^{5} + 86 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 107 T^{2} - 14 T^{3} + 7882 T^{4} - 14 p T^{5} - 107 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 3 T - 113 T^{2} + 36 T^{3} + 9792 T^{4} + 36 p T^{5} - 113 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 9 T + 164 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T - 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 19 T + 107 T^{2} - 1444 T^{3} + 24466 T^{4} - 1444 p T^{5} + 107 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 7 T - 29 T^{2} + 812 T^{3} - 8078 T^{4} + 812 p T^{5} - 29 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.69675799879268460658796123634, −6.52234710593637300417237023694, −6.07952497016127975895791974655, −6.07456118542516564663981529265, −6.05842059470533065658400473252, −5.74438790962042197793056864932, −5.44380021352831010331254476975, −5.36243694993734310049620528198, −5.13410643810228361426802515514, −4.57348439745308300746798002619, −4.56764184793435272466297460973, −4.17012792851454133035675836127, −3.83183166030962414472027756745, −3.76069167920619765663690776411, −3.60633255523269079437272511480, −3.28254181149445120485908300953, −2.93823598035367074995142850478, −2.74333738071149507649510473713, −2.58451573962371584225282479067, −2.12732362578530929528344508307, −1.92670211967839181431227630220, −1.56494678241911347016437634997, −1.47551615743428101748592828330, −0.77621079495670323449665891843, −0.68780073689650834684624357754,
0.68780073689650834684624357754, 0.77621079495670323449665891843, 1.47551615743428101748592828330, 1.56494678241911347016437634997, 1.92670211967839181431227630220, 2.12732362578530929528344508307, 2.58451573962371584225282479067, 2.74333738071149507649510473713, 2.93823598035367074995142850478, 3.28254181149445120485908300953, 3.60633255523269079437272511480, 3.76069167920619765663690776411, 3.83183166030962414472027756745, 4.17012792851454133035675836127, 4.56764184793435272466297460973, 4.57348439745308300746798002619, 5.13410643810228361426802515514, 5.36243694993734310049620528198, 5.44380021352831010331254476975, 5.74438790962042197793056864932, 6.05842059470533065658400473252, 6.07456118542516564663981529265, 6.07952497016127975895791974655, 6.52234710593637300417237023694, 6.69675799879268460658796123634
