| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6595653566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6595653566\) |
| \(L(1)\) |
|
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
−11.17796587863549209637293452690, −10.48961318140679390867228266638, −9.079900869336769123594498457421, −8.269452619320731637465393114544, −7.41648394388131976267690221315, −6.33913726419224037569636207263, −5.43776788456016476213579825651, −4.43264569837597764632094150272, −3.44536275764821128051758051565, −1.27727610380497217722898197237,
1.27727610380497217722898197237, 3.44536275764821128051758051565, 4.43264569837597764632094150272, 5.43776788456016476213579825651, 6.33913726419224037569636207263, 7.41648394388131976267690221315, 8.269452619320731637465393114544, 9.079900869336769123594498457421, 10.48961318140679390867228266638, 11.17796587863549209637293452690
