| L(s) = 1 | + 4-s − 2·7-s − 8·13-s + 16-s + 4·19-s − 25-s − 2·28-s + 10·31-s + 4·37-s − 20·43-s − 11·49-s − 8·52-s + 16·61-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 16·79-s + 16·91-s − 2·97-s − 100-s − 8·103-s + 4·109-s − 2·112-s − 13·121-s + 10·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 1/5·25-s − 0.377·28-s + 1.79·31-s + 0.657·37-s − 3.04·43-s − 1.57·49-s − 1.10·52-s + 2.04·61-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 1.80·79-s + 1.67·91-s − 0.203·97-s − 0.0999·100-s − 0.788·103-s + 0.383·109-s − 0.188·112-s − 1.18·121-s + 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7229881866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7229881866\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−12.98699917725805927719758339046, −11.98990591115535383596132902997, −11.91184860007480462763991644584, −11.21947207780374695340874205378, −10.11526477822370291644410044895, −9.872045258839871807563899217385, −9.512741326522107701681952133776, −8.268048909391662018895990050852, −7.82723843094092867969941089213, −6.73860900674537002760989556093, −6.71508725988629390136097161445, −5.37363015537505503550836847886, −4.77709333112371596512170551343, −3.38549178369413487864105189864, −2.44850978333263328730546301919,
2.44850978333263328730546301919, 3.38549178369413487864105189864, 4.77709333112371596512170551343, 5.37363015537505503550836847886, 6.71508725988629390136097161445, 6.73860900674537002760989556093, 7.82723843094092867969941089213, 8.268048909391662018895990050852, 9.512741326522107701681952133776, 9.872045258839871807563899217385, 10.11526477822370291644410044895, 11.21947207780374695340874205378, 11.91184860007480462763991644584, 11.98990591115535383596132902997, 12.98699917725805927719758339046
