| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 3·9-s + 2·11-s + 4·12-s − 4·16-s + 4·17-s + 6·18-s + 12·19-s + 4·22-s − 9·25-s + 4·27-s − 8·32-s + 4·33-s + 8·34-s + 6·36-s + 24·38-s + 20·41-s − 20·43-s + 4·44-s − 8·48-s − 5·49-s − 18·50-s + 8·51-s + 8·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 9-s + 0.603·11-s + 1.15·12-s − 16-s + 0.970·17-s + 1.41·18-s + 2.75·19-s + 0.852·22-s − 9/5·25-s + 0.769·27-s − 1.41·32-s + 0.696·33-s + 1.37·34-s + 36-s + 3.89·38-s + 3.12·41-s − 3.04·43-s + 0.603·44-s − 1.15·48-s − 5/7·49-s − 2.54·50-s + 1.12·51-s + 1.08·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.236466889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.236466889\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−7.85729183798579044278610418571, −7.52539857035144694302700552711, −7.25489076239522002618528148628, −6.46487870715497229983731599323, −6.32235586317594684409084519104, −5.59473778933342030293105926599, −5.14885571821443324588162204875, −5.05726722612368152650778432824, −4.08682412119350237454079486362, −3.77257881491072191084851286374, −3.55072107725385727672342073940, −2.93657054326743540839942565323, −2.51282770938376169099813859692, −1.74132409908467826753927107484, −1.01286372155151662703812038439,
1.01286372155151662703812038439, 1.74132409908467826753927107484, 2.51282770938376169099813859692, 2.93657054326743540839942565323, 3.55072107725385727672342073940, 3.77257881491072191084851286374, 4.08682412119350237454079486362, 5.05726722612368152650778432824, 5.14885571821443324588162204875, 5.59473778933342030293105926599, 6.32235586317594684409084519104, 6.46487870715497229983731599323, 7.25489076239522002618528148628, 7.52539857035144694302700552711, 7.85729183798579044278610418571
