| L(s) = 1 | − 2-s + 4-s − 0.692·5-s − 0.356·7-s − 8-s + 0.692·10-s + 2.93·11-s + 0.356·14-s + 16-s − 6.71·17-s + 7.20·19-s − 0.692·20-s − 2.93·22-s − 2.39·23-s − 4.52·25-s − 0.356·28-s − 7.82·29-s + 2.76·31-s − 32-s + 6.71·34-s + 0.246·35-s − 10.0·37-s − 7.20·38-s + 0.692·40-s + 4.89·41-s + 6.59·43-s + 2.93·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.309·5-s − 0.134·7-s − 0.353·8-s + 0.218·10-s + 0.886·11-s + 0.0953·14-s + 0.250·16-s − 1.62·17-s + 1.65·19-s − 0.154·20-s − 0.626·22-s − 0.499·23-s − 0.904·25-s − 0.0674·28-s − 1.45·29-s + 0.496·31-s − 0.176·32-s + 1.15·34-s + 0.0417·35-s − 1.66·37-s − 1.16·38-s + 0.109·40-s + 0.763·41-s + 1.00·43-s + 0.443·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.692T + 5T^{2} \) |
| 7 | \( 1 + 0.356T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 - 1.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 6.81T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 0.396T + 89T^{2} \) |
| 97 | \( 1 - 0.417T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.423069041930001195071718742599, −7.54494654803559069286065657438, −7.05483972315288101485203920033, −6.20396114467422436656260008460, −5.42805428604844832595547951041, −4.24841263267134275871458242060, −3.56251804291097970523003834742, −2.40420703742196743148810854159, −1.38923057390762519928632329771, 0,
1.38923057390762519928632329771, 2.40420703742196743148810854159, 3.56251804291097970523003834742, 4.24841263267134275871458242060, 5.42805428604844832595547951041, 6.20396114467422436656260008460, 7.05483972315288101485203920033, 7.54494654803559069286065657438, 8.423069041930001195071718742599
