| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s − 15-s − 16-s + 3·17-s + 3·19-s − 4·23-s − 24-s + 27-s + 30-s − 2·31-s − 3·34-s − 3·38-s + 40-s + 4·46-s + 48-s + 49-s − 3·51-s − 54-s − 3·57-s + 2·62-s + 64-s + 4·69-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s − 15-s − 16-s + 3·17-s + 3·19-s − 4·23-s − 24-s + 27-s + 30-s − 2·31-s − 3·34-s − 3·38-s + 40-s + 4·46-s + 48-s + 49-s − 3·51-s − 54-s − 3·57-s + 2·62-s + 64-s + 4·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5380416639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5380416639\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.760719117157179335477536961089, −9.293895101378969554992896468928, −9.200623121079635895436243915718, −8.280470741270518240845199245697, −7.999766181447529397573491630855, −7.67085084270521232180187820740, −7.58534180535230040184065763639, −6.93564146566920443849925101270, −6.38356022435968803072346121589, −5.82802862488768338832935354203, −5.51745900886662951846251860828, −5.38008321369507625796634825797, −5.26777616187360311551577507386, −4.08162247933992172967937618909, −3.95156007953557878507315166732, −3.28882891341080085674290845811, −2.73156890467627738618637121648, −1.72939569467481382069176179786, −1.56923718534117202579117379907, −0.74359213069873547407954506489,
0.74359213069873547407954506489, 1.56923718534117202579117379907, 1.72939569467481382069176179786, 2.73156890467627738618637121648, 3.28882891341080085674290845811, 3.95156007953557878507315166732, 4.08162247933992172967937618909, 5.26777616187360311551577507386, 5.38008321369507625796634825797, 5.51745900886662951846251860828, 5.82802862488768338832935354203, 6.38356022435968803072346121589, 6.93564146566920443849925101270, 7.58534180535230040184065763639, 7.67085084270521232180187820740, 7.999766181447529397573491630855, 8.280470741270518240845199245697, 9.200623121079635895436243915718, 9.293895101378969554992896468928, 9.760719117157179335477536961089
