| L(s) = 1 | + 2-s + 0.732·3-s + 4-s − 1.73·5-s + 0.732·6-s − 2·7-s + 8-s − 2.46·9-s − 1.73·10-s + 1.26·11-s + 0.732·12-s + 3.46·13-s − 2·14-s − 1.26·15-s + 16-s − 4.26·17-s − 2.46·18-s + 4.73·19-s − 1.73·20-s − 1.46·21-s + 1.26·22-s − 1.26·23-s + 0.732·24-s − 2.00·25-s + 3.46·26-s − 4·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s − 0.774·5-s + 0.298·6-s − 0.755·7-s + 0.353·8-s − 0.821·9-s − 0.547·10-s + 0.382·11-s + 0.211·12-s + 0.960·13-s − 0.534·14-s − 0.327·15-s + 0.250·16-s − 1.03·17-s − 0.580·18-s + 1.08·19-s − 0.387·20-s − 0.319·21-s + 0.270·22-s − 0.264·23-s + 0.149·24-s − 0.400·25-s + 0.679·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 8.66T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 + 0.196T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 4.26T + 97T^{2} \) |
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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
−8.440409622199163947794741119380, −7.60637182309198619999902546440, −6.91047436276092639395768863924, −6.03396961650184837807967465929, −5.44044354987971704583589709051, −4.19213866662575127366344050155, −3.59838178842967269665208327861, −3.00411989387278110851113694780, −1.77623580547409915280527186107, 0,
1.77623580547409915280527186107, 3.00411989387278110851113694780, 3.59838178842967269665208327861, 4.19213866662575127366344050155, 5.44044354987971704583589709051, 6.03396961650184837807967465929, 6.91047436276092639395768863924, 7.60637182309198619999902546440, 8.440409622199163947794741119380
