| L(s) = 1 | + 3-s − 4·4-s − 2·9-s − 4·12-s + 12·16-s − 25-s − 5·27-s + 10·31-s + 8·36-s − 14·37-s + 12·48-s + 14·49-s − 32·64-s − 26·67-s − 75-s + 81-s + 10·93-s + 34·97-s + 4·100-s − 8·103-s + 20·108-s − 14·111-s − 11·121-s − 40·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s − 2/3·9-s − 1.15·12-s + 3·16-s − 1/5·25-s − 0.962·27-s + 1.79·31-s + 4/3·36-s − 2.30·37-s + 1.73·48-s + 2·49-s − 4·64-s − 3.17·67-s − 0.115·75-s + 1/9·81-s + 1.03·93-s + 3.45·97-s + 2/5·100-s − 0.788·103-s + 1.92·108-s − 1.32·111-s − 121-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4460885105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4460885105\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−17.17113961041653810289302193238, −16.85574838471113963425433979456, −15.74352154365985779192189445348, −15.15355040523506862802895652520, −14.48010355709446529915257600066, −14.04167765341748268011552322659, −13.52133580755889374274588661744, −13.25737799010645723889294192930, −12.18377057708802742812991424989, −11.91285142562008747901103505623, −10.50315208485772406638675869216, −10.14183943032400540896029256073, −9.160132383755029723980325860329, −8.841889407311114617556569512502, −8.264947596125555144370168258865, −7.48410396012403277235033293890, −6.01987084005356094215710711834, −5.18130267836277930636751198273, −4.27630887656717266868263341494, −3.25699087747366984275342802206,
3.25699087747366984275342802206, 4.27630887656717266868263341494, 5.18130267836277930636751198273, 6.01987084005356094215710711834, 7.48410396012403277235033293890, 8.264947596125555144370168258865, 8.841889407311114617556569512502, 9.160132383755029723980325860329, 10.14183943032400540896029256073, 10.50315208485772406638675869216, 11.91285142562008747901103505623, 12.18377057708802742812991424989, 13.25737799010645723889294192930, 13.52133580755889374274588661744, 14.04167765341748268011552322659, 14.48010355709446529915257600066, 15.15355040523506862802895652520, 15.74352154365985779192189445348, 16.85574838471113963425433979456, 17.17113961041653810289302193238
