| L(s) = 1 | + 7-s + 2·13-s + 2·19-s − 6·29-s − 8·31-s − 4·37-s + 6·41-s − 2·43-s − 6·47-s + 49-s + 6·53-s + 12·59-s − 8·61-s − 2·67-s − 6·71-s − 2·73-s + 16·79-s + 6·89-s + 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.554·13-s + 0.458·19-s − 1.11·29-s − 1.43·31-s − 0.657·37-s + 0.937·41-s − 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 1.02·61-s − 0.244·67-s − 0.712·71-s − 0.234·73-s + 1.80·79-s + 0.635·89-s + 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.05037448605997, −13.31233158445123, −13.19958802746968, −12.54520378122353, −12.00567878204389, −11.43638642080959, −11.18021144281442, −10.53199648593850, −10.17892752722173, −9.408828719748676, −9.049386868341896, −8.627217587595941, −7.874058233397042, −7.553510606217893, −7.020350756459550, −6.363516259114635, −5.823347328372593, −5.270695190259510, −4.878203925562348, −3.958060529154987, −3.703671688195259, −3.002599099055786, −2.173699545317300, −1.673154926991259, −0.9220105164690834, 0,
0.9220105164690834, 1.673154926991259, 2.173699545317300, 3.002599099055786, 3.703671688195259, 3.958060529154987, 4.878203925562348, 5.270695190259510, 5.823347328372593, 6.363516259114635, 7.020350756459550, 7.553510606217893, 7.874058233397042, 8.627217587595941, 9.049386868341896, 9.408828719748676, 10.17892752722173, 10.53199648593850, 11.18021144281442, 11.43638642080959, 12.00567878204389, 12.54520378122353, 13.19958802746968, 13.31233158445123, 14.05037448605997
