| L(s) = 1 | − 2·3-s − 5-s + 9-s + 11-s − 10·13-s + 2·15-s − 8·17-s − 5·19-s − 8·23-s − 4·25-s + 2·27-s + 6·29-s − 2·31-s − 2·33-s − 3·37-s + 20·39-s − 12·41-s − 14·43-s − 45-s − 12·47-s + 16·51-s − 11·53-s − 55-s + 10·57-s + 5·59-s + 20·61-s + 10·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 2.77·13-s + 0.516·15-s − 1.94·17-s − 1.14·19-s − 1.66·23-s − 4/5·25-s + 0.384·27-s + 1.11·29-s − 0.359·31-s − 0.348·33-s − 0.493·37-s + 3.20·39-s − 1.87·41-s − 2.13·43-s − 0.149·45-s − 1.75·47-s + 2.24·51-s − 1.51·53-s − 0.134·55-s + 1.32·57-s + 0.650·59-s + 2.56·61-s + 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09758923010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09758923010\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 7 T^{2} + 14 T^{3} - 68 T^{4} + 14 p T^{5} - 7 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T - 5 T^{2} - 40 T^{3} + 64 T^{4} - 40 p T^{5} - 5 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 36 p T^{5} - 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 176 p T^{5} - p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 40 p T^{5} - 85 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 14 p T^{5} - 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + T - 131 T^{2} - 14 T^{3} + 12022 T^{4} - 14 p T^{5} - 131 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 328 p T^{5} - 53 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 94 T^{2} - 288 T^{3} + 5775 T^{4} - 288 p T^{5} - 94 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.81695554299455741788409266176, −6.80512260048271359462295050699, −6.79758672154317782143429289794, −6.42428770262686765925234944122, −6.14626560195690827268689998232, −5.77763686053225862967228062558, −5.77744173490622074560342659634, −5.22271490222358933032557084005, −5.04420104956756392718015910702, −5.01946102642159717060802657711, −4.98323915875044978100695559708, −4.37908555838206050748524471014, −4.32085149648720455609570160613, −4.04417137946267281504634604849, −3.91720846801189983315946156300, −3.58890387015977182604098150945, −3.05193540861414762761767418945, −2.74653797183982337296454501122, −2.73033687499624013240957497978, −2.27959762688732794083328690386, −1.87961689359995519293193694671, −1.70143797118374783547436750214, −1.47958167855104934382636022283, −0.29436549027709707823040348888, −0.19940388449532264378368721982,
0.19940388449532264378368721982, 0.29436549027709707823040348888, 1.47958167855104934382636022283, 1.70143797118374783547436750214, 1.87961689359995519293193694671, 2.27959762688732794083328690386, 2.73033687499624013240957497978, 2.74653797183982337296454501122, 3.05193540861414762761767418945, 3.58890387015977182604098150945, 3.91720846801189983315946156300, 4.04417137946267281504634604849, 4.32085149648720455609570160613, 4.37908555838206050748524471014, 4.98323915875044978100695559708, 5.01946102642159717060802657711, 5.04420104956756392718015910702, 5.22271490222358933032557084005, 5.77744173490622074560342659634, 5.77763686053225862967228062558, 6.14626560195690827268689998232, 6.42428770262686765925234944122, 6.79758672154317782143429289794, 6.80512260048271359462295050699, 6.81695554299455741788409266176
