| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s + 4·6-s − 8·8-s − 5·9-s + 14·11-s − 8·12-s + 16·16-s + 2·17-s + 10·18-s − 34·19-s − 28·22-s + 16·24-s + 25·25-s + 28·27-s − 32·32-s − 28·33-s − 4·34-s − 20·36-s + 68·38-s − 46·41-s + 14·43-s + 56·44-s − 32·48-s + 49·49-s − 50·50-s + ⋯ |
| L(s) = 1 | − 2-s − 2/3·3-s + 4-s + 2/3·6-s − 8-s − 5/9·9-s + 1.27·11-s − 2/3·12-s + 16-s + 2/17·17-s + 5/9·18-s − 1.78·19-s − 1.27·22-s + 2/3·24-s + 25-s + 1.03·27-s − 32-s − 0.848·33-s − 0.117·34-s − 5/9·36-s + 1.78·38-s − 1.12·41-s + 0.325·43-s + 1.27·44-s − 2/3·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3886892575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3886892575\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{2} T^{2} \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 14 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 2 T + p^{2} T^{2} \) |
| 19 | \( 1 + 34 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 + 46 T + p^{2} T^{2} \) |
| 43 | \( 1 - 14 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( 1 + 82 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 62 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 142 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 158 T + p^{2} T^{2} \) |
| 89 | \( 1 - 146 T + p^{2} T^{2} \) |
| 97 | \( 1 + 94 T + p^{2} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.87048677617965783923537238867, −20.22348755982430789264039507830, −18.98117139949859357867964680426, −17.39390564234539719905896023939, −16.66238633994296888505128348462, −14.78865148952925327223857635554, −12.10310589791838852466797231478, −10.76188679715007134885201539453, −8.799074388015237098898553961892, −6.45219985833962822825614349363,
6.45219985833962822825614349363, 8.799074388015237098898553961892, 10.76188679715007134885201539453, 12.10310589791838852466797231478, 14.78865148952925327223857635554, 16.66238633994296888505128348462, 17.39390564234539719905896023939, 18.98117139949859357867964680426, 20.22348755982430789264039507830, 21.87048677617965783923537238867
