| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s + 4·11-s − 2·12-s + 16-s − 4·17-s − 3·18-s + 4·22-s − 2·24-s − 9·25-s + 14·27-s + 32-s − 8·33-s − 4·34-s − 3·36-s + 4·41-s + 8·43-s + 4·44-s − 2·48-s − 5·49-s − 9·50-s + 8·51-s + 14·54-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s + 1.20·11-s − 0.577·12-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.852·22-s − 0.408·24-s − 9/5·25-s + 2.69·27-s + 0.176·32-s − 1.39·33-s − 0.685·34-s − 1/2·36-s + 0.624·41-s + 1.21·43-s + 0.603·44-s − 0.288·48-s − 5/7·49-s − 1.27·50-s + 1.12·51-s + 1.90·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−7.900574670861696081136405729725, −7.50453196645377754362608030446, −6.87081593462957102969561941235, −6.48988831247255432885204202586, −6.11586748167354289065343430906, −5.84304457304729519676851015132, −5.34006059048988750874758197539, −4.95996492835075275892547561988, −4.13073034219794476961344305330, −4.08090613362146692342747948187, −3.28231565080656425004767856193, −2.60362431459742126306839445746, −2.06983106012671431074676520675, −1.06485830879781620251017267814, 0,
1.06485830879781620251017267814, 2.06983106012671431074676520675, 2.60362431459742126306839445746, 3.28231565080656425004767856193, 4.08090613362146692342747948187, 4.13073034219794476961344305330, 4.95996492835075275892547561988, 5.34006059048988750874758197539, 5.84304457304729519676851015132, 6.11586748167354289065343430906, 6.48988831247255432885204202586, 6.87081593462957102969561941235, 7.50453196645377754362608030446, 7.900574670861696081136405729725
