L(s) = 1 | + 2-s − 2.95·3-s + 4-s − 5-s − 2.95·6-s + 7-s + 8-s + 5.71·9-s − 10-s − 2.95·12-s + 5.80·13-s + 14-s + 2.95·15-s + 16-s − 0.469·17-s + 5.71·18-s − 4.04·19-s − 20-s − 2.95·21-s − 7.69·23-s − 2.95·24-s + 25-s + 5.80·26-s − 8.00·27-s + 28-s + 2.63·29-s + 2.95·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.70·3-s + 0.5·4-s − 0.447·5-s − 1.20·6-s + 0.377·7-s + 0.353·8-s + 1.90·9-s − 0.316·10-s − 0.852·12-s + 1.60·13-s + 0.267·14-s + 0.762·15-s + 0.250·16-s − 0.113·17-s + 1.34·18-s − 0.928·19-s − 0.223·20-s − 0.644·21-s − 1.60·23-s − 0.602·24-s + 0.200·25-s + 1.13·26-s − 1.54·27-s + 0.188·28-s + 0.488·29-s + 0.538·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.95T + 3T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 17 | \( 1 + 0.469T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 0.790T + 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 3.95T + 79T^{2} \) |
| 83 | \( 1 + 9.44T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.08419850544799107467397852415, −6.53771657260069842374633438365, −5.91877879399534447124142931040, −5.54088013190963640715571558342, −4.58467318685541233387182275856, −4.16509218884189341552647559557, −3.45154993010908848864252007163, −2.02786599958810357895669064796, −1.18191539225703612469118377861, 0,
1.18191539225703612469118377861, 2.02786599958810357895669064796, 3.45154993010908848864252007163, 4.16509218884189341552647559557, 4.58467318685541233387182275856, 5.54088013190963640715571558342, 5.91877879399534447124142931040, 6.53771657260069842374633438365, 7.08419850544799107467397852415