L(s) = 1 | − 2·3-s − 7-s + 9-s + 4·11-s + 6·13-s + 2·21-s + 23-s + 4·27-s − 2·29-s − 2·31-s − 8·33-s + 2·37-s − 12·39-s − 6·41-s − 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s − 12·61-s − 63-s + 4·67-s − 2·69-s + 2·73-s − 4·77-s + 8·79-s − 11·81-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 0.371·29-s − 0.359·31-s − 1.39·33-s + 0.328·37-s − 1.92·39-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s − 1.53·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 0.234·73-s − 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.12544233644331, −12.46375799375848, −11.99760576115676, −11.65435905314851, −11.25500334317636, −10.89284960360497, −10.38570607176451, −9.938128790551428, −9.303250872920562, −8.889187258683773, −8.481642369206326, −7.982636271502178, −7.153727281792670, −6.648439109732981, −6.535878441934546, −5.896237698914970, −5.518534798474819, −5.076985511856857, −4.259488908761391, −3.881003163357587, −3.445485533219894, −2.761650265502228, −1.885108344993052, −1.250842509597153, −0.8242180484062006, 0,
0.8242180484062006, 1.250842509597153, 1.885108344993052, 2.761650265502228, 3.445485533219894, 3.881003163357587, 4.259488908761391, 5.076985511856857, 5.518534798474819, 5.896237698914970, 6.535878441934546, 6.648439109732981, 7.153727281792670, 7.982636271502178, 8.481642369206326, 8.889187258683773, 9.303250872920562, 9.938128790551428, 10.38570607176451, 10.89284960360497, 11.25500334317636, 11.65435905314851, 11.99760576115676, 12.46375799375848, 13.12544233644331