| L(s) = 1 | − 3·4-s + 2·7-s + 4·13-s + 4·16-s + 28·19-s + 3·25-s − 6·28-s − 6·31-s − 12·37-s − 16·43-s + 49-s − 12·52-s + 16·61-s − 9·64-s + 4·67-s − 84·76-s + 8·79-s + 8·91-s + 24·97-s − 9·100-s + 26·103-s + 36·109-s + 8·112-s + 15·121-s + 18·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.755·7-s + 1.10·13-s + 16-s + 6.42·19-s + 3/5·25-s − 1.13·28-s − 1.07·31-s − 1.97·37-s − 2.43·43-s + 1/7·49-s − 1.66·52-s + 2.04·61-s − 9/8·64-s + 0.488·67-s − 9.63·76-s + 0.900·79-s + 0.838·91-s + 2.43·97-s − 0.899·100-s + 2.56·103-s + 3.44·109-s + 0.755·112-s + 1.36·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876839776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876839776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 3 T^{2} - 16 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 15 T^{2} + 104 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 17 T^{2} - 240 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 30 T^{2} + 59 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 75 T^{2} + 3944 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 86 T^{2} + 507 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 165 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.69679198748802826983559658472, −7.50931943475277156915929184087, −7.29406964875944349046920551781, −7.17430625861009816682813996908, −6.84347978611362862299762395545, −6.82795187870317546873145463931, −6.12140467144691533590486421017, −5.88626963209246009090443632817, −5.70955801295667906090318073610, −5.42585368077153635625500422181, −5.19369002616297607518684244554, −5.14753551535851252444716110218, −4.84283726508459786850027281540, −4.51728902817707379647895443485, −4.49384331146462366318721638305, −3.65264254732891797011775210284, −3.44361010995422362270805818530, −3.43460000464808613195212431072, −3.20829445335908572224602172141, −3.15808075737164071060245866665, −2.06866229084575812617631073718, −1.97695338729716011605404330780, −1.34722206450387431720409302345, −0.940417049047063968904108612557, −0.75660365681407308590198330003,
0.75660365681407308590198330003, 0.940417049047063968904108612557, 1.34722206450387431720409302345, 1.97695338729716011605404330780, 2.06866229084575812617631073718, 3.15808075737164071060245866665, 3.20829445335908572224602172141, 3.43460000464808613195212431072, 3.44361010995422362270805818530, 3.65264254732891797011775210284, 4.49384331146462366318721638305, 4.51728902817707379647895443485, 4.84283726508459786850027281540, 5.14753551535851252444716110218, 5.19369002616297607518684244554, 5.42585368077153635625500422181, 5.70955801295667906090318073610, 5.88626963209246009090443632817, 6.12140467144691533590486421017, 6.82795187870317546873145463931, 6.84347978611362862299762395545, 7.17430625861009816682813996908, 7.29406964875944349046920551781, 7.50931943475277156915929184087, 7.69679198748802826983559658472
