| L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s − 6·11-s + 2·14-s + 5·16-s + 12·22-s − 12·23-s − 25-s − 3·28-s + 12·29-s − 6·32-s + 4·37-s − 20·43-s − 18·44-s + 24·46-s − 6·49-s + 2·50-s + 18·53-s + 4·56-s − 24·58-s + 7·64-s + 28·67-s − 8·74-s + 6·77-s + 16·79-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s − 1.80·11-s + 0.534·14-s + 5/4·16-s + 2.55·22-s − 2.50·23-s − 1/5·25-s − 0.566·28-s + 2.22·29-s − 1.06·32-s + 0.657·37-s − 3.04·43-s − 2.71·44-s + 3.53·46-s − 6/7·49-s + 0.282·50-s + 2.47·53-s + 0.534·56-s − 3.15·58-s + 7/8·64-s + 3.42·67-s − 0.929·74-s + 0.683·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4733068702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4733068702\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.533580822054688698380792321620, −8.613282561215628799056639387984, −8.268048909391662018895990050852, −8.073998889221963720660118720673, −7.69509009195539198448319901638, −6.73860900674537002760989556093, −6.69870395284666594287906434089, −5.97811569986712198414996682793, −5.37363015537505503550836847886, −4.86220558498840854064591879984, −3.91930425992199330786644833836, −3.20132525085319240177530071259, −2.44850978333263328730546301919, −1.98881454498950312538243430831, −0.55647773141554817914458641913,
0.55647773141554817914458641913, 1.98881454498950312538243430831, 2.44850978333263328730546301919, 3.20132525085319240177530071259, 3.91930425992199330786644833836, 4.86220558498840854064591879984, 5.37363015537505503550836847886, 5.97811569986712198414996682793, 6.69870395284666594287906434089, 6.73860900674537002760989556093, 7.69509009195539198448319901638, 8.073998889221963720660118720673, 8.268048909391662018895990050852, 8.613282561215628799056639387984, 9.533580822054688698380792321620
