L(s) = 1 | − 4·7-s − 4·11-s − 4·13-s + 4·17-s − 4·19-s − 8·23-s − 6·29-s + 4·31-s − 2·37-s − 10·41-s − 43-s − 4·47-s + 9·49-s − 2·53-s − 12·59-s + 8·67-s − 12·71-s + 14·73-s + 16·77-s − 8·79-s − 6·83-s + 4·89-s + 16·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.20·11-s − 1.10·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s − 0.152·43-s − 0.583·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s + 0.977·67-s − 1.42·71-s + 1.63·73-s + 1.82·77-s − 0.900·79-s − 0.658·83-s + 0.423·89-s + 1.67·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.59847625849663, −12.95891720830240, −12.62214592436205, −12.20211420214165, −11.88536829927738, −11.08023962402138, −10.53375118205618, −10.04846531090456, −9.832966574261676, −9.486386972765865, −8.678639187764243, −8.129627189892817, −7.798713548013766, −7.147841573001395, −6.754548978323673, −6.078656865597327, −5.755618734667706, −5.160178795231655, −4.563734976665062, −3.949600491533822, −3.187324592308352, −3.040850936038882, −2.172327826738090, −1.778098853775560, −0.4615164200356730, 0,
0.4615164200356730, 1.778098853775560, 2.172327826738090, 3.040850936038882, 3.187324592308352, 3.949600491533822, 4.563734976665062, 5.160178795231655, 5.755618734667706, 6.078656865597327, 6.754548978323673, 7.147841573001395, 7.798713548013766, 8.129627189892817, 8.678639187764243, 9.486386972765865, 9.832966574261676, 10.04846531090456, 10.53375118205618, 11.08023962402138, 11.88536829927738, 12.20211420214165, 12.62214592436205, 12.95891720830240, 13.59847625849663