| L(s) = 1 | + 2-s − 2.68·3-s + 4-s + 0.326·5-s − 2.68·6-s + 2.00·7-s + 8-s + 4.18·9-s + 0.326·10-s + 2.08·11-s − 2.68·12-s − 6.30·13-s + 2.00·14-s − 0.876·15-s + 16-s + 1.13·17-s + 4.18·18-s − 3.28·19-s + 0.326·20-s − 5.37·21-s + 2.08·22-s + 2.66·23-s − 2.68·24-s − 4.89·25-s − 6.30·26-s − 3.19·27-s + 2.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.54·3-s + 0.5·4-s + 0.146·5-s − 1.09·6-s + 0.757·7-s + 0.353·8-s + 1.39·9-s + 0.103·10-s + 0.627·11-s − 0.774·12-s − 1.74·13-s + 0.535·14-s − 0.226·15-s + 0.250·16-s + 0.276·17-s + 0.987·18-s − 0.754·19-s + 0.0730·20-s − 1.17·21-s + 0.444·22-s + 0.555·23-s − 0.547·24-s − 0.978·25-s − 1.23·26-s − 0.613·27-s + 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 0.326T + 5T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 3.28T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 + 2.23T + 83T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.82627237419854447364091070480, −6.89558838426924401853132667508, −6.67906110162478166089367826285, −5.55076667974022359550309382606, −5.18079544684157736665818786490, −4.59803829670090917544260361106, −3.72247712314273466041215986342, −2.35852682575920701542531808246, −1.43404661879440272411315515370, 0,
1.43404661879440272411315515370, 2.35852682575920701542531808246, 3.72247712314273466041215986342, 4.59803829670090917544260361106, 5.18079544684157736665818786490, 5.55076667974022359550309382606, 6.67906110162478166089367826285, 6.89558838426924401853132667508, 7.82627237419854447364091070480
