| L(s) = 1 | − 2-s + 4-s + 6·5-s − 8-s − 6·10-s − 8·13-s + 16-s + 6·20-s + 17·25-s + 8·26-s + 12·29-s − 32-s + 4·37-s − 6·40-s − 12·41-s − 13·49-s − 17·50-s − 8·52-s + 18·53-s − 12·58-s + 16·61-s + 64-s − 48·65-s − 14·73-s − 4·74-s + 6·80-s + 12·82-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 1.89·10-s − 2.21·13-s + 1/4·16-s + 1.34·20-s + 17/5·25-s + 1.56·26-s + 2.22·29-s − 0.176·32-s + 0.657·37-s − 0.948·40-s − 1.87·41-s − 1.85·49-s − 2.40·50-s − 1.10·52-s + 2.47·53-s − 1.57·58-s + 2.04·61-s + 1/8·64-s − 5.95·65-s − 1.63·73-s − 0.464·74-s + 0.670·80-s + 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.252252272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252252272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−10.13919133269169682063036547080, −10.11526477822370291644410044895, −9.872045258839871807563899217385, −9.405159791372798554846122664326, −8.657482823158305382635077732253, −8.268048909391662018895990050852, −7.29929175763847648434390847974, −6.73860900674537002760989556093, −6.43760550649778711412630154056, −5.43287815412844995352588032205, −5.37363015537505503550836847886, −4.47356450228848094997188248522, −2.78595567768432254175608596610, −2.44850978333263328730546301919, −1.56286118008502235583804201463,
1.56286118008502235583804201463, 2.44850978333263328730546301919, 2.78595567768432254175608596610, 4.47356450228848094997188248522, 5.37363015537505503550836847886, 5.43287815412844995352588032205, 6.43760550649778711412630154056, 6.73860900674537002760989556093, 7.29929175763847648434390847974, 8.268048909391662018895990050852, 8.657482823158305382635077732253, 9.405159791372798554846122664326, 9.872045258839871807563899217385, 10.11526477822370291644410044895, 10.13919133269169682063036547080
