| L(s) = 1 | + 3-s − 5-s + 7-s + 11-s + 13-s − 15-s − 17-s + 21-s + 25-s − 27-s + 29-s + 33-s − 35-s + 39-s − 47-s + 49-s − 51-s − 55-s − 65-s + 2·71-s + 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s + 85-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 7-s + 11-s + 13-s − 15-s − 17-s + 21-s + 25-s − 27-s + 29-s + 33-s − 35-s + 39-s − 47-s + 49-s − 51-s − 55-s − 65-s + 2·71-s + 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s + 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.609556686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609556686\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.820738877963536642368942996802, −8.500227427684979222680892116017, −7.974148538596621093813012882005, −7.04287686447460118284552543387, −6.28106336659729637455684868216, −5.01962788217033758359746052772, −4.14183443191368691548019728093, −3.59232776293508694724433552002, −2.51374964405774669040363009954, −1.33514658438003097784799390782,
1.33514658438003097784799390782, 2.51374964405774669040363009954, 3.59232776293508694724433552002, 4.14183443191368691548019728093, 5.01962788217033758359746052772, 6.28106336659729637455684868216, 7.04287686447460118284552543387, 7.974148538596621093813012882005, 8.500227427684979222680892116017, 8.820738877963536642368942996802
