| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 1.45·7-s − 8-s + 9-s + 10-s − 3.50·11-s − 12-s + 4.95·13-s + 1.45·14-s + 15-s + 16-s − 3.19·17-s − 18-s + 3.19·19-s − 20-s + 1.45·21-s + 3.50·22-s + 4.15·23-s + 24-s + 25-s − 4.95·26-s − 27-s − 1.45·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.548·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.05·11-s − 0.288·12-s + 1.37·13-s + 0.387·14-s + 0.258·15-s + 0.250·16-s − 0.774·17-s − 0.235·18-s + 0.732·19-s − 0.223·20-s + 0.316·21-s + 0.748·22-s + 0.866·23-s + 0.204·24-s + 0.200·25-s − 0.972·26-s − 0.192·27-s − 0.274·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 - 0.900T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 0.549T + 61T^{2} \) |
| 71 | \( 1 + 6.64T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−8.673598346568477288345290025951, −8.104182060500285290746785738297, −7.17238684189321177053864607109, −6.53600082656102606678507907787, −5.71029150942541205664539627689, −4.80684426009838869038709551529, −3.63867274015453187377595458156, −2.74432168020512818391406893252, −1.26908834560176971307462856784, 0,
1.26908834560176971307462856784, 2.74432168020512818391406893252, 3.63867274015453187377595458156, 4.80684426009838869038709551529, 5.71029150942541205664539627689, 6.53600082656102606678507907787, 7.17238684189321177053864607109, 8.104182060500285290746785738297, 8.673598346568477288345290025951
