| L(s) = 1 | − 9-s − 4·11-s − 4·19-s − 12·29-s − 4·31-s − 20·41-s − 49-s + 16·59-s − 4·61-s − 20·71-s + 81-s − 4·89-s + 4·99-s − 12·101-s + 28·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 4·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.20·11-s − 0.917·19-s − 2.22·29-s − 0.718·31-s − 3.12·41-s − 1/7·49-s + 2.08·59-s − 0.512·61-s − 2.37·71-s + 1/9·81-s − 0.423·89-s + 0.402·99-s − 1.19·101-s + 2.68·109-s − 0.909·121-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 + 0.305·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1917249278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1917249278\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.301937215550944071328515025940, −8.621618392614239873815125769025, −8.577242328406466319352978760858, −8.383656814787692730998199955707, −7.57368372945139556058656651491, −7.36220778775770456810332207816, −7.20627459405208267087787105295, −6.39179153417117371018808781593, −6.27914087527691853401387882290, −5.49725271017642337833106955652, −5.46204864132805778753100852409, −4.97708332450268880609183935520, −4.50309858388751131446906466727, −3.84165440582506064484745163952, −3.56450321801565425881537584698, −3.03461871075041817737562240181, −2.41788727084920350811030816656, −1.97217767069043706481230098516, −1.44825648772511851006022703499, −0.14720698211932610698994654897,
0.14720698211932610698994654897, 1.44825648772511851006022703499, 1.97217767069043706481230098516, 2.41788727084920350811030816656, 3.03461871075041817737562240181, 3.56450321801565425881537584698, 3.84165440582506064484745163952, 4.50309858388751131446906466727, 4.97708332450268880609183935520, 5.46204864132805778753100852409, 5.49725271017642337833106955652, 6.27914087527691853401387882290, 6.39179153417117371018808781593, 7.20627459405208267087787105295, 7.36220778775770456810332207816, 7.57368372945139556058656651491, 8.383656814787692730998199955707, 8.577242328406466319352978760858, 8.621618392614239873815125769025, 9.301937215550944071328515025940
