| L(s) = 1 | − 3-s − 4-s + 2·8-s − 2·9-s − 11-s + 12-s − 8·13-s + 16-s + 3·17-s + 8·23-s − 2·24-s − 6·25-s + 2·27-s + 12·29-s + 4·31-s − 4·32-s + 33-s + 2·36-s + 4·37-s + 8·39-s − 12·41-s + 8·43-s + 44-s − 12·47-s − 48-s + 2·49-s − 3·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.707·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 0.727·17-s + 1.66·23-s − 0.408·24-s − 6/5·25-s + 0.384·27-s + 2.22·29-s + 0.718·31-s − 0.707·32-s + 0.174·33-s + 1/3·36-s + 0.657·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s + 0.150·44-s − 1.75·47-s − 0.144·48-s + 2/7·49-s − 0.420·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4262949369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4262949369\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.9164486365, −19.0798046347, −19.0177567201, −17.7512408724, −17.6761697408, −16.9473294095, −16.7727116030, −16.0129129656, −15.0693535709, −14.7742712184, −13.9795262067, −13.5527745379, −12.7606431103, −11.9362311444, −11.8560024471, −10.7173409989, −10.1733530745, −9.62624986142, −8.69568671187, −7.81910395524, −7.24765397307, −6.15685276164, −5.03220048416, −4.74199315541, −2.90847219286,
2.90847219286, 4.74199315541, 5.03220048416, 6.15685276164, 7.24765397307, 7.81910395524, 8.69568671187, 9.62624986142, 10.1733530745, 10.7173409989, 11.8560024471, 11.9362311444, 12.7606431103, 13.5527745379, 13.9795262067, 14.7742712184, 15.0693535709, 16.0129129656, 16.7727116030, 16.9473294095, 17.6761697408, 17.7512408724, 19.0177567201, 19.0798046347, 19.9164486365
