| L(s) = 1 | + 4-s − 3·16-s + 8·19-s + 16·31-s − 14·49-s + 4·61-s − 7·64-s + 8·76-s + 32·79-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 + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3/4·16-s + 1.83·19-s + 2.87·31-s − 2·49-s + 0.512·61-s − 7/8·64-s + 0.917·76-s + 3.60·79-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 + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575014416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575014416\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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
−12.24585028390759671322413107395, −11.84398909630952371043332939468, −11.74690789160613403410368998186, −11.05294965077240898604675348140, −10.65259519130546393506529691558, −10.04198369538576760343525987751, −9.427181850360461378258420474759, −9.397005801557129385093236827331, −8.361638436707407037404732372865, −8.058574380337454448787468535229, −7.54293240984768091665500147012, −6.76301274587898694338444462086, −6.55428878104586497175311860571, −5.88368349574315693264548254986, −5.01672023245190115764283599751, −4.75691278882402142163935293995, −3.76993668551353217309528527282, −3.02477398985994833328267227345, −2.39073745840140963186525318063, −1.17988259436947085351712516811,
1.17988259436947085351712516811, 2.39073745840140963186525318063, 3.02477398985994833328267227345, 3.76993668551353217309528527282, 4.75691278882402142163935293995, 5.01672023245190115764283599751, 5.88368349574315693264548254986, 6.55428878104586497175311860571, 6.76301274587898694338444462086, 7.54293240984768091665500147012, 8.058574380337454448787468535229, 8.361638436707407037404732372865, 9.397005801557129385093236827331, 9.427181850360461378258420474759, 10.04198369538576760343525987751, 10.65259519130546393506529691558, 11.05294965077240898604675348140, 11.74690789160613403410368998186, 11.84398909630952371043332939468, 12.24585028390759671322413107395
