L(s) = 1 | − 7-s − 6·11-s + 2·13-s + 2·17-s + 4·19-s − 4·23-s + 2·29-s + 2·31-s − 10·37-s − 6·41-s + 2·43-s + 2·47-s + 49-s + 6·53-s + 4·59-s + 12·61-s − 10·67-s + 12·71-s + 2·73-s + 6·77-s + 16·79-s + 12·83-s − 14·89-s − 2·91-s − 18·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 0.371·29-s + 0.359·31-s − 1.64·37-s − 0.937·41-s + 0.304·43-s + 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.53·61-s − 1.22·67-s + 1.42·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s + 1.31·83-s − 1.48·89-s − 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.78774726548900, −13.61317594944813, −13.08426378693197, −12.49804602560414, −12.06607762599689, −11.68638784287625, −10.82572247894190, −10.60814782088973, −10.04123989615840, −9.726575477287270, −9.022789670901755, −8.363387157700990, −8.057793338595005, −7.572065750911044, −6.896292109638312, −6.499830601568692, −5.652080788646434, −5.323520033228944, −4.993829526770624, −3.960558509418045, −3.617163088656679, −2.856998790102373, −2.453824875447625, −1.634844397108997, −0.7853413712040585, 0,
0.7853413712040585, 1.634844397108997, 2.453824875447625, 2.856998790102373, 3.617163088656679, 3.960558509418045, 4.993829526770624, 5.323520033228944, 5.652080788646434, 6.499830601568692, 6.896292109638312, 7.572065750911044, 8.057793338595005, 8.363387157700990, 9.022789670901755, 9.726575477287270, 10.04123989615840, 10.60814782088973, 10.82572247894190, 11.68638784287625, 12.06607762599689, 12.49804602560414, 13.08426378693197, 13.61317594944813, 13.78774726548900