| L(s) = 1 | − 4-s − 3·16-s − 8·19-s − 5·25-s + 16·31-s + 14·49-s + 4·61-s + 7·64-s + 8·76-s − 32·79-s + 5·100-s + 28·109-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3/4·16-s − 1.83·19-s − 25-s + 2.87·31-s + 2·49-s + 0.512·61-s + 7/8·64-s + 0.917·76-s − 3.60·79-s + 1/2·100-s + 2.68·109-s − 2·121-s − 1.43·124-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5869732169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5869732169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−16.04425621665513637432335881101, −15.39860978254693558799803182116, −15.31086366215355683397844908499, −14.18964599311887806872946724336, −14.03829554572186610771548305855, −13.15723379959748243640876161274, −12.96574043565923018826779476430, −11.92843380880703408712681539674, −11.66876266699699984465493727952, −10.71802248997556069743590768513, −10.18166894925267983021659200493, −9.570850536528072839705758819793, −8.529513714199935388900375216623, −8.506199327349381512983870230050, −7.36447877954871795658687378464, −6.52566376180141557577636495203, −5.84873109781954041807523169102, −4.63076666346438307078059690242, −4.10424053282794606423576741067, −2.49439517148849264400889437608,
2.49439517148849264400889437608, 4.10424053282794606423576741067, 4.63076666346438307078059690242, 5.84873109781954041807523169102, 6.52566376180141557577636495203, 7.36447877954871795658687378464, 8.506199327349381512983870230050, 8.529513714199935388900375216623, 9.570850536528072839705758819793, 10.18166894925267983021659200493, 10.71802248997556069743590768513, 11.66876266699699984465493727952, 11.92843380880703408712681539674, 12.96574043565923018826779476430, 13.15723379959748243640876161274, 14.03829554572186610771548305855, 14.18964599311887806872946724336, 15.31086366215355683397844908499, 15.39860978254693558799803182116, 16.04425621665513637432335881101
