L(s) = 1 | − 2·3-s − 5-s − 6·7-s + 9-s − 11-s − 10·13-s + 2·15-s + 8·17-s − 5·19-s + 12·21-s − 8·23-s − 4·25-s + 2·27-s − 6·29-s + 2·31-s + 2·33-s + 6·35-s + 3·37-s + 20·39-s + 12·41-s + 14·43-s − 45-s + 12·47-s + 13·49-s − 16·51-s + 11·53-s + 55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s − 2.26·7-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 + 2.61·21-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 + 1.01·35-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 + 13/7·49-s − 2.24·51-s + 1.51·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5115289143\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5115289143\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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
−6.90974073369889466858273688766, −6.37906705044863851482617974735, −6.30192465828549041339079319791, −6.23131943020615218500253017556, −6.13134634249591319064980477679, −5.78695323371799730126126090951, −5.58375431155042638305665284624, −5.25781589845314368053699112841, −5.24725512063074691394726526676, −5.04684595788311813563322490976, −4.65657431717327280099050646353, −4.21852907591029181225428672969, −4.06830084517847150167052716824, −3.88009072373067929237243309922, −3.85233798314375505081059322936, −3.44048312551222311982981539421, −3.14326378523093708816207401085, −2.57536464287427881807438246442, −2.52105586965396797450994190197, −2.41210971315156519291778211640, −2.24490970831535102301269291865, −1.59459120329683756991026790421, −0.805934704748375686104086349315, −0.62176364172155249081991722309, −0.29865888675088596953876190110,
0.29865888675088596953876190110, 0.62176364172155249081991722309, 0.805934704748375686104086349315, 1.59459120329683756991026790421, 2.24490970831535102301269291865, 2.41210971315156519291778211640, 2.52105586965396797450994190197, 2.57536464287427881807438246442, 3.14326378523093708816207401085, 3.44048312551222311982981539421, 3.85233798314375505081059322936, 3.88009072373067929237243309922, 4.06830084517847150167052716824, 4.21852907591029181225428672969, 4.65657431717327280099050646353, 5.04684595788311813563322490976, 5.24725512063074691394726526676, 5.25781589845314368053699112841, 5.58375431155042638305665284624, 5.78695323371799730126126090951, 6.13134634249591319064980477679, 6.23131943020615218500253017556, 6.30192465828549041339079319791, 6.37906705044863851482617974735, 6.90974073369889466858273688766