L(s) = 1 | − 2.22·2-s − 3-s + 2.97·4-s + 3.25·5-s + 2.22·6-s − 0.940·7-s − 2.16·8-s + 9-s − 7.26·10-s − 11-s − 2.97·12-s + 1.71·13-s + 2.09·14-s − 3.25·15-s − 1.11·16-s − 0.0942·17-s − 2.22·18-s + 3.20·19-s + 9.67·20-s + 0.940·21-s + 2.22·22-s − 6.07·23-s + 2.16·24-s + 5.60·25-s − 3.83·26-s − 27-s − 2.79·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.48·4-s + 1.45·5-s + 0.910·6-s − 0.355·7-s − 0.764·8-s + 0.333·9-s − 2.29·10-s − 0.301·11-s − 0.857·12-s + 0.476·13-s + 0.560·14-s − 0.840·15-s − 0.279·16-s − 0.0228·17-s − 0.525·18-s + 0.736·19-s + 2.16·20-s + 0.205·21-s + 0.475·22-s − 1.26·23-s + 0.441·24-s + 1.12·25-s − 0.751·26-s − 0.192·27-s − 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 0.940T + 7T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 + 0.0942T + 17T^{2} \) |
| 19 | \( 1 - 3.20T + 19T^{2} \) |
| 23 | \( 1 + 6.07T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 0.677T + 59T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 - 7.54T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 - 0.268T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−9.015940272679333387519033720585, −8.123873399762931812160094556058, −7.25740868788825576474029954579, −6.51535559147462371652077524422, −5.81110639075662039044084399954, −5.08141021939397257645916184185, −3.52363614168463623910592590331, −2.10956160673091595855996549406, −1.49387267041741172161357617891, 0,
1.49387267041741172161357617891, 2.10956160673091595855996549406, 3.52363614168463623910592590331, 5.08141021939397257645916184185, 5.81110639075662039044084399954, 6.51535559147462371652077524422, 7.25740868788825576474029954579, 8.123873399762931812160094556058, 9.015940272679333387519033720585