| L(s) = 1 | + 2·5-s + 2·17-s + 3·25-s + 2·37-s − 2·49-s − 2·53-s + 2·61-s + 2·73-s + 4·85-s − 2·89-s + 2·101-s − 2·109-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·17-s + 3·25-s + 2·37-s − 2·49-s − 2·53-s + 2·61-s + 2·73-s + 4·85-s − 2·89-s + 2·101-s − 2·109-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.569873684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569873684\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71098384564245410073796613728, −6.27856565477865577395681115374, −6.24111988969012474158524434410, −6.07750704687062031585181274565, −5.93452839075905779406593191283, −5.75563409448769321394720435853, −5.39704011476411240567593763831, −5.25701013261298193427637008779, −5.14426014108578891396799577458, −4.79300881731443883999701089725, −4.65752590781038788970274637330, −4.50837949496145981223053116923, −4.19756026004570153111352147272, −3.59826379470201509323266010958, −3.58886916400583979528153467484, −3.49844705769664769310765060146, −3.11120316724633613472369783783, −2.77853052213191372413307267916, −2.58832097175427601810190195703, −2.42231405924243423694679105265, −2.01786355941077241713527930167, −1.78806040064603763143423482924, −1.41251267060910699789832089450, −1.05388797508579704550789810452, −0.977336590523553512391287653304,
0.977336590523553512391287653304, 1.05388797508579704550789810452, 1.41251267060910699789832089450, 1.78806040064603763143423482924, 2.01786355941077241713527930167, 2.42231405924243423694679105265, 2.58832097175427601810190195703, 2.77853052213191372413307267916, 3.11120316724633613472369783783, 3.49844705769664769310765060146, 3.58886916400583979528153467484, 3.59826379470201509323266010958, 4.19756026004570153111352147272, 4.50837949496145981223053116923, 4.65752590781038788970274637330, 4.79300881731443883999701089725, 5.14426014108578891396799577458, 5.25701013261298193427637008779, 5.39704011476411240567593763831, 5.75563409448769321394720435853, 5.93452839075905779406593191283, 6.07750704687062031585181274565, 6.24111988969012474158524434410, 6.27856565477865577395681115374, 6.71098384564245410073796613728
