| L(s) = 1 | − 1.83·2-s + 2.22·3-s + 1.36·4-s + 1.24·5-s − 4.07·6-s + 2.01·7-s + 1.16·8-s + 1.93·9-s − 2.27·10-s − 5.64·11-s + 3.02·12-s − 2.77·13-s − 3.68·14-s + 2.76·15-s − 4.86·16-s + 0.668·17-s − 3.54·18-s + 4.24·19-s + 1.69·20-s + 4.46·21-s + 10.3·22-s − 4.21·23-s + 2.59·24-s − 3.45·25-s + 5.09·26-s − 2.36·27-s + 2.74·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 1.28·3-s + 0.681·4-s + 0.555·5-s − 1.66·6-s + 0.759·7-s + 0.412·8-s + 0.644·9-s − 0.720·10-s − 1.70·11-s + 0.874·12-s − 0.770·13-s − 0.985·14-s + 0.712·15-s − 1.21·16-s + 0.162·17-s − 0.836·18-s + 0.973·19-s + 0.378·20-s + 0.974·21-s + 2.20·22-s − 0.879·23-s + 0.529·24-s − 0.690·25-s + 0.999·26-s − 0.455·27-s + 0.517·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 0.668T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + 2.51T + 47T^{2} \) |
| 53 | \( 1 - 9.20T + 53T^{2} \) |
| 59 | \( 1 + 8.56T + 59T^{2} \) |
| 61 | \( 1 + 9.37T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 0.728T + 71T^{2} \) |
| 73 | \( 1 + 9.77T + 73T^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.074942483544563441609249995025, −7.61330363470290214596540578891, −7.36006908865639819455425904890, −5.81619702832283240139567945323, −5.13976644418678465107111757713, −4.18768520829296449517738892290, −2.94360988345965009764765559307, −2.25642962707815948861937773283, −1.59358013164298890836999157734, 0,
1.59358013164298890836999157734, 2.25642962707815948861937773283, 2.94360988345965009764765559307, 4.18768520829296449517738892290, 5.13976644418678465107111757713, 5.81619702832283240139567945323, 7.36006908865639819455425904890, 7.61330363470290214596540578891, 8.074942483544563441609249995025
