| L(s) = 1 | − 2·2-s + 4·3-s + 4-s − 8·6-s + 8·7-s + 2·8-s + 6·9-s + 4·12-s − 12·13-s − 16·14-s − 4·16-s + 12·17-s − 12·18-s − 2·19-s + 32·21-s + 8·24-s − 8·25-s + 24·26-s − 4·27-s + 8·28-s + 2·32-s − 24·34-s + 6·36-s + 4·38-s − 48·39-s + 6·41-s − 64·42-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s − 3.26·6-s + 3.02·7-s + 0.707·8-s + 2·9-s + 1.15·12-s − 3.32·13-s − 4.27·14-s − 16-s + 2.91·17-s − 2.82·18-s − 0.458·19-s + 6.98·21-s + 1.63·24-s − 8/5·25-s + 4.70·26-s − 0.769·27-s + 1.51·28-s + 0.353·32-s − 4.11·34-s + 36-s + 0.648·38-s − 7.68·39-s + 0.937·41-s − 9.87·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.193698164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193698164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 507 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 456 T^{3} + 1971 T^{4} - 456 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3810 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 14 T^{2} + 64 T^{3} + 1483 T^{4} + 64 p T^{5} - 14 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 56 T^{2} + 96 T^{3} + 111 T^{4} + 96 p T^{5} + 56 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 26 T^{2} - 144 T^{3} + 3483 T^{4} - 144 p T^{5} + 26 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 131 T^{2} + 1350 T^{3} + 14652 T^{4} + 1350 p T^{5} + 131 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} - 80 T^{3} + 9799 T^{4} - 80 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 714 T^{3} + 10476 T^{4} + 714 p T^{5} + 131 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T - 47 T^{2} + 190 T^{3} - 3020 T^{4} + 190 p T^{5} - 47 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 86 T^{2} + 1155 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 278 T^{2} + 2880 T^{3} + 29619 T^{4} + 2880 p T^{5} + 278 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2682 T^{3} + 23268 T^{4} + 2682 p T^{5} + 257 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.840858066176326772679822925622, −9.531196663781359117394746118593, −9.487398685572345572993080132627, −9.312890924620304496582503963563, −8.726384060449015888927563690374, −8.633204051433608548029331777161, −8.112689184043381657184008671386, −8.042270286817653117544590641933, −7.928640561208158473309675077217, −7.75173649679101607519044460897, −7.36690078948811546224439219263, −7.34105964881883823157273450015, −7.00914468353950443903852884434, −5.95679877052569384345613007563, −5.75397282070503442631068894885, −5.24990632257554510788593133287, −5.12257764103634430479328663228, −4.65012206407404933622648897726, −4.23139950921729108461214450473, −4.05455537546136089266178135595, −3.21534186766200558312398054343, −2.82772192678536030945170781749, −2.41441984541700376014803708653, −1.71825552365282122115149657755, −1.68465876703108235790528508369,
1.68465876703108235790528508369, 1.71825552365282122115149657755, 2.41441984541700376014803708653, 2.82772192678536030945170781749, 3.21534186766200558312398054343, 4.05455537546136089266178135595, 4.23139950921729108461214450473, 4.65012206407404933622648897726, 5.12257764103634430479328663228, 5.24990632257554510788593133287, 5.75397282070503442631068894885, 5.95679877052569384345613007563, 7.00914468353950443903852884434, 7.34105964881883823157273450015, 7.36690078948811546224439219263, 7.75173649679101607519044460897, 7.928640561208158473309675077217, 8.042270286817653117544590641933, 8.112689184043381657184008671386, 8.633204051433608548029331777161, 8.726384060449015888927563690374, 9.312890924620304496582503963563, 9.487398685572345572993080132627, 9.531196663781359117394746118593, 9.840858066176326772679822925622
