| L(s) = 1 | + 1.64·2-s − 3-s + 0.709·4-s − 5-s − 1.64·6-s + 5.13·7-s − 2.12·8-s + 9-s − 1.64·10-s + 4.21·11-s − 0.709·12-s − 6.37·13-s + 8.44·14-s + 15-s − 4.91·16-s − 6.62·17-s + 1.64·18-s + 5.24·19-s − 0.709·20-s − 5.13·21-s + 6.93·22-s − 2.30·23-s + 2.12·24-s + 25-s − 10.4·26-s − 27-s + 3.64·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.354·4-s − 0.447·5-s − 0.671·6-s + 1.94·7-s − 0.751·8-s + 0.333·9-s − 0.520·10-s + 1.27·11-s − 0.204·12-s − 1.76·13-s + 2.25·14-s + 0.258·15-s − 1.22·16-s − 1.60·17-s + 0.387·18-s + 1.20·19-s − 0.158·20-s − 1.12·21-s + 1.47·22-s − 0.480·23-s + 0.433·24-s + 0.200·25-s − 2.05·26-s − 0.192·27-s + 0.688·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 2.28T + 29T^{2} \) |
| 31 | \( 1 + 6.36T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 0.660T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 1.92T + 61T^{2} \) |
| 67 | \( 1 - 5.01T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 + 0.255T + 97T^{2} \) |
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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
−7.47941335192616689363429562988, −6.98282636724648938752915061112, −6.11640582494180983809557263294, −5.18450346517348924131551498470, −4.83236247291133659537774874578, −4.35676918261167826138763180494, −3.59239975455824296460593509006, −2.34921713805422832862191504365, −1.53059049303468164890475443673, 0,
1.53059049303468164890475443673, 2.34921713805422832862191504365, 3.59239975455824296460593509006, 4.35676918261167826138763180494, 4.83236247291133659537774874578, 5.18450346517348924131551498470, 6.11640582494180983809557263294, 6.98282636724648938752915061112, 7.47941335192616689363429562988
