L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s + 13-s − 3·17-s + 2·19-s + 21-s + 3·23-s − 27-s − 6·29-s − 2·31-s − 3·33-s − 7·37-s − 39-s + 9·41-s − 8·43-s − 6·47-s − 6·49-s + 3·51-s + 3·53-s − 2·57-s + 7·61-s − 63-s + 4·67-s − 3·69-s − 3·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 0.625·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.522·33-s − 1.15·37-s − 0.160·39-s + 1.40·41-s − 1.21·43-s − 0.875·47-s − 6/7·49-s + 0.420·51-s + 0.412·53-s − 0.264·57-s + 0.896·61-s − 0.125·63-s + 0.488·67-s − 0.361·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.56645705139779, −14.03738668448252, −13.32841241791099, −13.04874658770488, −12.54832660762646, −11.86111980632215, −11.53625032673253, −10.98172300628512, −10.64737679248400, −9.733166872436571, −9.574615896857947, −8.905005523639052, −8.449292399600114, −7.685501093867190, −7.041966349321920, −6.713084797568384, −6.157645103833254, −5.564128886519707, −4.984483016695678, −4.410356312923326, −3.599118648247027, −3.378746200555967, −2.287466333560774, −1.654717396253110, −0.8810866421153675, 0,
0.8810866421153675, 1.654717396253110, 2.287466333560774, 3.378746200555967, 3.599118648247027, 4.410356312923326, 4.984483016695678, 5.564128886519707, 6.157645103833254, 6.713084797568384, 7.041966349321920, 7.685501093867190, 8.449292399600114, 8.905005523639052, 9.574615896857947, 9.733166872436571, 10.64737679248400, 10.98172300628512, 11.53625032673253, 11.86111980632215, 12.54832660762646, 13.04874658770488, 13.32841241791099, 14.03738668448252, 14.56645705139779