L(s) = 1 | − 0.865·2-s + 3-s − 1.25·4-s + 1.66·5-s − 0.865·6-s − 0.715·7-s + 2.81·8-s + 9-s − 1.44·10-s − 0.656·11-s − 1.25·12-s − 4.37·13-s + 0.619·14-s + 1.66·15-s + 0.0635·16-s − 7.15·17-s − 0.865·18-s + 6.56·19-s − 2.08·20-s − 0.715·21-s + 0.568·22-s − 23-s + 2.81·24-s − 2.22·25-s + 3.78·26-s + 27-s + 0.895·28-s + ⋯ |
L(s) = 1 | − 0.612·2-s + 0.577·3-s − 0.625·4-s + 0.744·5-s − 0.353·6-s − 0.270·7-s + 0.995·8-s + 0.333·9-s − 0.456·10-s − 0.198·11-s − 0.360·12-s − 1.21·13-s + 0.165·14-s + 0.430·15-s + 0.0158·16-s − 1.73·17-s − 0.204·18-s + 1.50·19-s − 0.465·20-s − 0.156·21-s + 0.121·22-s − 0.208·23-s + 0.574·24-s − 0.445·25-s + 0.742·26-s + 0.192·27-s + 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 0.865T + 2T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 + 0.715T + 7T^{2} \) |
| 11 | \( 1 + 0.656T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 31 | \( 1 - 0.705T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 8.50T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 - 0.253T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 0.611T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.904704381263216884158021627486, −8.173414260767610638602659962862, −7.32349218803630308320014115936, −6.66688470030188812727642744517, −5.36974594319357538081360019351, −4.79619832863514215636766824474, −3.72574072111506657014122739012, −2.56902408649765190830487222917, −1.63836418820017179196332876904, 0,
1.63836418820017179196332876904, 2.56902408649765190830487222917, 3.72574072111506657014122739012, 4.79619832863514215636766824474, 5.36974594319357538081360019351, 6.66688470030188812727642744517, 7.32349218803630308320014115936, 8.173414260767610638602659962862, 8.904704381263216884158021627486