| L(s) = 1 | − 2-s − 1.63·3-s + 4-s + 1.19·5-s + 1.63·6-s + 2.04·7-s − 8-s − 0.314·9-s − 1.19·10-s − 3.68·11-s − 1.63·12-s + 2.15·13-s − 2.04·14-s − 1.95·15-s + 16-s + 6.53·17-s + 0.314·18-s − 5.31·19-s + 1.19·20-s − 3.34·21-s + 3.68·22-s − 5.89·23-s + 1.63·24-s − 3.57·25-s − 2.15·26-s + 5.43·27-s + 2.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.946·3-s + 0.5·4-s + 0.533·5-s + 0.669·6-s + 0.772·7-s − 0.353·8-s − 0.104·9-s − 0.377·10-s − 1.11·11-s − 0.473·12-s + 0.598·13-s − 0.546·14-s − 0.505·15-s + 0.250·16-s + 1.58·17-s + 0.0740·18-s − 1.21·19-s + 0.266·20-s − 0.730·21-s + 0.785·22-s − 1.22·23-s + 0.334·24-s − 0.714·25-s − 0.422·26-s + 1.04·27-s + 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 3.68T + 43T^{2} \) |
| 47 | \( 1 - 0.992T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.05T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
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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
−8.944344313237129851681446292578, −8.040610119978549353972865892501, −7.64080535092400146134386375029, −6.35859890000377535222527739171, −5.71694738123485296165326669110, −5.24095476361463182909966960255, −3.91720070203683245323678309126, −2.50885852630779814634927688259, −1.47593652820112294085842230992, 0,
1.47593652820112294085842230992, 2.50885852630779814634927688259, 3.91720070203683245323678309126, 5.24095476361463182909966960255, 5.71694738123485296165326669110, 6.35859890000377535222527739171, 7.64080535092400146134386375029, 8.040610119978549353972865892501, 8.944344313237129851681446292578
