L(s) = 1 | − 7-s + 2·11-s − 4·13-s + 6·17-s − 6·19-s + 8·23-s + 2·29-s − 10·31-s − 2·37-s − 10·41-s − 4·43-s + 8·47-s + 49-s + 4·53-s − 8·59-s + 6·61-s + 12·67-s − 6·71-s + 12·73-s − 2·77-s + 8·79-s + 4·83-s + 10·89-s + 4·91-s − 8·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.603·11-s − 1.10·13-s + 1.45·17-s − 1.37·19-s + 1.66·23-s + 0.371·29-s − 1.79·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s − 1.04·59-s + 0.768·61-s + 1.46·67-s − 0.712·71-s + 1.40·73-s − 0.227·77-s + 0.900·79-s + 0.439·83-s + 1.05·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.40449273717015, −15.03389731609236, −14.65923442467843, −14.12208146235760, −13.46370129084106, −12.81122959572959, −12.40805932923096, −12.02749351637510, −11.28594725111076, −10.64571411562821, −10.24691225883780, −9.516313577771579, −9.148947316041449, −8.495741466934525, −7.850454613868564, −7.076612438577563, −6.846777189285347, −6.080412304202392, −5.193978630149594, −5.025757839090358, −3.928242706718422, −3.499509586250852, −2.696466296394976, −1.938382035049136, −1.035122734538915, 0,
1.035122734538915, 1.938382035049136, 2.696466296394976, 3.499509586250852, 3.928242706718422, 5.025757839090358, 5.193978630149594, 6.080412304202392, 6.846777189285347, 7.076612438577563, 7.850454613868564, 8.495741466934525, 9.148947316041449, 9.516313577771579, 10.24691225883780, 10.64571411562821, 11.28594725111076, 12.02749351637510, 12.40805932923096, 12.81122959572959, 13.46370129084106, 14.12208146235760, 14.65923442467843, 15.03389731609236, 15.40449273717015