L(s) = 1 | + 5-s − 7-s − 2·11-s + 4·13-s − 6·17-s + 6·19-s + 8·23-s + 25-s + 2·29-s + 10·31-s − 35-s + 2·37-s − 10·41-s − 4·43-s + 8·47-s + 49-s − 4·53-s − 2·55-s + 8·59-s + 6·61-s + 4·65-s + 12·67-s + 6·71-s − 12·73-s + 2·77-s − 8·79-s + 4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s + 1.10·13-s − 1.45·17-s + 1.37·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s + 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.549·53-s − 0.269·55-s + 1.04·59-s + 0.768·61-s + 0.496·65-s + 1.46·67-s + 0.712·71-s − 1.40·73-s + 0.227·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770649992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770649992\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.696452714310980436804590274048, −8.851474858651842534028322626469, −8.247604606498974890796864988463, −7.02157725521702622435483532726, −6.48456250977973837758319022992, −5.45966473542222361729415904781, −4.66829124920132920735782458172, −3.40332551255394100686318391218, −2.51987438160165085152083981910, −1.03137336342293280097139560363,
1.03137336342293280097139560363, 2.51987438160165085152083981910, 3.40332551255394100686318391218, 4.66829124920132920735782458172, 5.45966473542222361729415904781, 6.48456250977973837758319022992, 7.02157725521702622435483532726, 8.247604606498974890796864988463, 8.851474858651842534028322626469, 9.696452714310980436804590274048