| L(s) = 1 | + 0.803·2-s − 3-s − 1.35·4-s − 0.904·5-s − 0.803·6-s + 1.61·7-s − 2.69·8-s + 9-s − 0.727·10-s − 1.65·11-s + 1.35·12-s − 0.0916·13-s + 1.29·14-s + 0.904·15-s + 0.539·16-s − 17-s + 0.803·18-s − 5.68·19-s + 1.22·20-s − 1.61·21-s − 1.32·22-s + 5.14·23-s + 2.69·24-s − 4.18·25-s − 0.0736·26-s − 27-s − 2.18·28-s + ⋯ |
| L(s) = 1 | + 0.568·2-s − 0.577·3-s − 0.676·4-s − 0.404·5-s − 0.328·6-s + 0.608·7-s − 0.953·8-s + 0.333·9-s − 0.230·10-s − 0.498·11-s + 0.390·12-s − 0.0254·13-s + 0.346·14-s + 0.233·15-s + 0.134·16-s − 0.242·17-s + 0.189·18-s − 1.30·19-s + 0.273·20-s − 0.351·21-s − 0.283·22-s + 1.07·23-s + 0.550·24-s − 0.836·25-s − 0.0144·26-s − 0.192·27-s − 0.412·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.803T + 2T^{2} \) |
| 5 | \( 1 + 0.904T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 + 0.0916T + 13T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 9.54T + 37T^{2} \) |
| 41 | \( 1 - 0.791T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 + 0.249T + 47T^{2} \) |
| 53 | \( 1 + 2.79T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 + 1.36T + 61T^{2} \) |
| 67 | \( 1 - 8.35T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 0.693T + 97T^{2} \) |
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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
−7.52386447869975581718186856827, −6.55130084527476953142914026741, −6.05209744939979817737748433114, −5.22224256051065541552566707004, −4.52375485628677203867027668530, −4.33824269321254699953058164603, −3.23704304808308747906956012274, −2.38548760613051141696104181157, −1.06141174369463085617705111212, 0,
1.06141174369463085617705111212, 2.38548760613051141696104181157, 3.23704304808308747906956012274, 4.33824269321254699953058164603, 4.52375485628677203867027668530, 5.22224256051065541552566707004, 6.05209744939979817737748433114, 6.55130084527476953142914026741, 7.52386447869975581718186856827
