| L(s) = 1 | + 2·3-s + 9-s + 8·13-s − 5·25-s − 4·27-s + 16·31-s − 16·37-s + 16·39-s − 4·43-s + 2·49-s − 4·67-s − 10·75-s + 16·79-s − 11·81-s + 32·93-s − 32·111-s + 8·117-s − 10·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 2.21·13-s − 25-s − 0.769·27-s + 2.87·31-s − 2.63·37-s + 2.56·39-s − 0.609·43-s + 2/7·49-s − 0.488·67-s − 1.15·75-s + 1.80·79-s − 1.22·81-s + 3.31·93-s − 3.03·111-s + 0.739·117-s − 0.909·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.329·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162990823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162990823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.961438715280076608971505144727, −9.356490597711443364839676768459, −8.823262242558443727697412393445, −8.402486721045359624563329453251, −8.224967449944354325923563142557, −7.61571802357545508290774757844, −6.81452531711109911039107167291, −6.34581089125020976995772108722, −5.84146207242263671711382935961, −5.08729499106778796759213281290, −4.25282051764710785820219227400, −3.59752127996725809703281146050, −3.24330744887432581245727497541, −2.29123964855756961770541680185, −1.36090134795392967233663774704,
1.36090134795392967233663774704, 2.29123964855756961770541680185, 3.24330744887432581245727497541, 3.59752127996725809703281146050, 4.25282051764710785820219227400, 5.08729499106778796759213281290, 5.84146207242263671711382935961, 6.34581089125020976995772108722, 6.81452531711109911039107167291, 7.61571802357545508290774757844, 8.224967449944354325923563142557, 8.402486721045359624563329453251, 8.823262242558443727697412393445, 9.356490597711443364839676768459, 9.961438715280076608971505144727
