| L(s) = 1 | + 2-s − 6·3-s + 4-s − 6·6-s + 8-s + 21·9-s − 6·12-s + 16-s + 21·18-s − 10·19-s − 6·24-s + 6·25-s − 54·27-s + 32-s + 21·36-s − 10·38-s + 8·41-s − 10·43-s − 6·48-s − 5·49-s + 6·50-s − 54·54-s + 60·57-s + 20·59-s + 64-s − 8·67-s + 21·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3.46·3-s + 1/2·4-s − 2.44·6-s + 0.353·8-s + 7·9-s − 1.73·12-s + 1/4·16-s + 4.94·18-s − 2.29·19-s − 1.22·24-s + 6/5·25-s − 10.3·27-s + 0.176·32-s + 7/2·36-s − 1.62·38-s + 1.24·41-s − 1.52·43-s − 0.866·48-s − 5/7·49-s + 0.848·50-s − 7.34·54-s + 7.94·57-s + 2.60·59-s + 1/8·64-s − 0.977·67-s + 2.47·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.015127034089896271738111260335, −7.19510826607411358550241547791, −6.79665889988949967719732090774, −6.72726504320701495197449588428, −6.06956565112627680353985311275, −6.00802937924645068372500171262, −5.31957522354665396045702186375, −5.04650939586985321436896693077, −4.59340547650806364343033240582, −4.22216049290866687245255073450, −3.69652118358615221384815805348, −2.53675892669785388180575829109, −1.69517936438262700861480943218, −0.891967053262301195338534272444, 0,
0.891967053262301195338534272444, 1.69517936438262700861480943218, 2.53675892669785388180575829109, 3.69652118358615221384815805348, 4.22216049290866687245255073450, 4.59340547650806364343033240582, 5.04650939586985321436896693077, 5.31957522354665396045702186375, 6.00802937924645068372500171262, 6.06956565112627680353985311275, 6.72726504320701495197449588428, 6.79665889988949967719732090774, 7.19510826607411358550241547791, 8.015127034089896271738111260335
