| L(s) = 1 | + 5·9-s − 6·11-s − 10·19-s − 4·31-s − 6·41-s + 10·49-s − 4·61-s − 24·71-s − 20·79-s + 16·81-s − 30·89-s − 30·99-s + 36·101-s − 20·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 50·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 1.80·11-s − 2.29·19-s − 0.718·31-s − 0.937·41-s + 10/7·49-s − 0.512·61-s − 2.84·71-s − 2.25·79-s + 16/9·81-s − 3.17·89-s − 3.01·99-s + 3.58·101-s − 1.91·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 3.82·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144359654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144359654\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.833473105060152715516327423133, −9.033129279835063836287112328697, −8.839874986700037821240829008085, −8.454802171153744946770242249215, −7.979946653368058210238291303002, −7.52265525667723835458661791976, −7.30643623631575389702948724994, −6.85912414369834048784166163008, −6.48520933267338993531911959906, −5.81944050046625790123303105400, −5.63412177363564889887943413343, −4.95642289359197992827615876786, −4.60083420206572865839383475998, −4.08701952912088618592638500784, −3.96626328181030401313763663332, −2.90176831911550508098756468628, −2.73991896352042329987197508111, −1.77825955755040677392653954616, −1.71065455482548057711580589783, −0.39734728218883045740931514747,
0.39734728218883045740931514747, 1.71065455482548057711580589783, 1.77825955755040677392653954616, 2.73991896352042329987197508111, 2.90176831911550508098756468628, 3.96626328181030401313763663332, 4.08701952912088618592638500784, 4.60083420206572865839383475998, 4.95642289359197992827615876786, 5.63412177363564889887943413343, 5.81944050046625790123303105400, 6.48520933267338993531911959906, 6.85912414369834048784166163008, 7.30643623631575389702948724994, 7.52265525667723835458661791976, 7.979946653368058210238291303002, 8.454802171153744946770242249215, 8.839874986700037821240829008085, 9.033129279835063836287112328697, 9.833473105060152715516327423133
