| L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 8-s + 9-s − 2·11-s − 16-s + 17-s − 18-s + 2·19-s + 2·22-s − 2·24-s + 2·27-s + 4·33-s − 34-s − 2·38-s + 41-s − 2·43-s + 2·48-s + 2·49-s − 2·51-s − 2·54-s − 4·57-s + 59-s + 64-s − 4·66-s + 67-s + ⋯ |
| L(s) = 1 | − 2-s − 2·3-s + 2·6-s + 8-s + 9-s − 2·11-s − 16-s + 17-s − 18-s + 2·19-s + 2·22-s − 2·24-s + 2·27-s + 4·33-s − 34-s − 2·38-s + 41-s − 2·43-s + 2·48-s + 2·49-s − 2·51-s − 2·54-s − 4·57-s + 59-s + 64-s − 4·66-s + 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2992783281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2992783281\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.122071345926588598575568097106, −8.332069598149599117981018553842, −8.204904509616653978322091610269, −7.66431842077474198234987457278, −7.45468440700127506174417905295, −7.21711840564239426605320191311, −6.60107986511674122415307284124, −6.24465723971237284711489609820, −5.72537649953636185351604359329, −5.45949140196912992993251475685, −5.27696142126478775552119289126, −4.83243154849529916787784962448, −4.77086111356691073240131427652, −3.86393011855442051595249922593, −3.39005638050539678256863861898, −2.87508101750869351776484650408, −2.43647482493826202488866773414, −1.68287882945680099695377238636, −0.76022739601535963651202880162, −0.68423752526969071470027603493,
0.68423752526969071470027603493, 0.76022739601535963651202880162, 1.68287882945680099695377238636, 2.43647482493826202488866773414, 2.87508101750869351776484650408, 3.39005638050539678256863861898, 3.86393011855442051595249922593, 4.77086111356691073240131427652, 4.83243154849529916787784962448, 5.27696142126478775552119289126, 5.45949140196912992993251475685, 5.72537649953636185351604359329, 6.24465723971237284711489609820, 6.60107986511674122415307284124, 7.21711840564239426605320191311, 7.45468440700127506174417905295, 7.66431842077474198234987457278, 8.204904509616653978322091610269, 8.332069598149599117981018553842, 9.122071345926588598575568097106
