L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 14-s + 16-s + 6·17-s − 8·19-s + 20-s + 25-s − 28-s − 6·29-s + 4·31-s − 32-s − 6·34-s − 35-s + 10·37-s + 8·38-s − 40-s − 6·41-s − 4·43-s + 49-s − 50-s + 6·53-s + 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.223·20-s + 1/5·25-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s + 1.64·37-s + 1.29·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.141·50-s + 0.824·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.75864340537045, −13.56068915969030, −12.81463969519722, −12.47386175242674, −12.04406448585310, −11.38923426499475, −10.83048424932188, −10.48079680131269, −9.967882462991365, −9.442464057028176, −9.199524477636392, −8.403888691800619, −8.032943633220067, −7.602769442433091, −6.873290890595641, −6.298465805137839, −6.118422296824051, −5.344602927956544, −4.820342523029125, −3.987867865753555, −3.503619253256403, −2.762567321676444, −2.218046276458991, −1.570272950974728, −0.8352540506298705, 0,
0.8352540506298705, 1.570272950974728, 2.218046276458991, 2.762567321676444, 3.503619253256403, 3.987867865753555, 4.820342523029125, 5.344602927956544, 6.118422296824051, 6.298465805137839, 6.873290890595641, 7.602769442433091, 8.032943633220067, 8.403888691800619, 9.199524477636392, 9.442464057028176, 9.967882462991365, 10.48079680131269, 10.83048424932188, 11.38923426499475, 12.04406448585310, 12.47386175242674, 12.81463969519722, 13.56068915969030, 13.75864340537045