| L(s) = 1 | + 2·3-s + 6·7-s − 9-s − 2·11-s − 6·17-s + 6·19-s + 12·21-s + 6·23-s − 2·25-s − 6·27-s + 6·29-s − 4·33-s − 2·37-s + 6·41-s + 6·43-s + 12·47-s + 15·49-s − 12·51-s + 12·57-s + 6·59-s + 14·61-s − 6·63-s + 18·67-s + 12·69-s − 6·71-s − 8·73-s − 4·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s − 1/3·9-s − 0.603·11-s − 1.45·17-s + 1.37·19-s + 2.61·21-s + 1.25·23-s − 2/5·25-s − 1.15·27-s + 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.937·41-s + 0.914·43-s + 1.75·47-s + 15/7·49-s − 1.68·51-s + 1.58·57-s + 0.781·59-s + 1.79·61-s − 0.755·63-s + 2.19·67-s + 1.44·69-s − 0.712·71-s − 0.936·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.227589939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.227589939\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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
−8.163002021581106903165300470532, −8.114993285590380313870147274227, −7.80260847911718824346767896848, −7.48768716252550825910324545140, −7.02670816124336143730025666275, −6.74815102104003767634057697284, −6.21158818162114220661713603136, −5.60232411800284578314719073859, −5.41867426713687951873961939835, −4.99160731173212377682020100826, −4.80414533850404788822905637947, −4.31453885098004132106142828463, −3.87484245944889210098693645298, −3.49568209900732479762338730748, −2.76205241891822037752605317653, −2.68208283939971958074764602206, −2.05182273091076588194428043791, −2.00228685098563934553756299625, −0.968780946651762960778269978958, −0.789429202330035618389570570369,
0.789429202330035618389570570369, 0.968780946651762960778269978958, 2.00228685098563934553756299625, 2.05182273091076588194428043791, 2.68208283939971958074764602206, 2.76205241891822037752605317653, 3.49568209900732479762338730748, 3.87484245944889210098693645298, 4.31453885098004132106142828463, 4.80414533850404788822905637947, 4.99160731173212377682020100826, 5.41867426713687951873961939835, 5.60232411800284578314719073859, 6.21158818162114220661713603136, 6.74815102104003767634057697284, 7.02670816124336143730025666275, 7.48768716252550825910324545140, 7.80260847911718824346767896848, 8.114993285590380313870147274227, 8.163002021581106903165300470532
