L(s) = 1 | − 2-s + 0.675·3-s + 4-s + 5-s − 0.675·6-s − 1.96·7-s − 8-s − 2.54·9-s − 10-s − 0.702·11-s + 0.675·12-s + 3.63·13-s + 1.96·14-s + 0.675·15-s + 16-s − 0.463·17-s + 2.54·18-s + 1.64·19-s + 20-s − 1.32·21-s + 0.702·22-s − 3.53·23-s − 0.675·24-s + 25-s − 3.63·26-s − 3.74·27-s − 1.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.390·3-s + 0.5·4-s + 0.447·5-s − 0.275·6-s − 0.743·7-s − 0.353·8-s − 0.847·9-s − 0.316·10-s − 0.211·11-s + 0.195·12-s + 1.00·13-s + 0.525·14-s + 0.174·15-s + 0.250·16-s − 0.112·17-s + 0.599·18-s + 0.376·19-s + 0.223·20-s − 0.289·21-s + 0.149·22-s − 0.736·23-s − 0.137·24-s + 0.200·25-s − 0.713·26-s − 0.720·27-s − 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 0.675T + 3T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 0.702T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 + 0.463T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 - 2.71T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.122318598879485376304102134084, −7.59788256847260969921965391570, −6.51725469135678926146204039111, −6.06554697067360742942212546251, −5.35486077454489779793914742262, −4.02869067176157377170357205750, −3.16206429490263436149127001606, −2.49561102606008902058867743273, −1.38714389737662786671511790870, 0,
1.38714389737662786671511790870, 2.49561102606008902058867743273, 3.16206429490263436149127001606, 4.02869067176157377170357205750, 5.35486077454489779793914742262, 6.06554697067360742942212546251, 6.51725469135678926146204039111, 7.59788256847260969921965391570, 8.122318598879485376304102134084