| L(s) = 1 | − 3-s + 3·7-s + 9-s − 3·11-s − 13-s − 3·17-s − 3·21-s + 3·23-s − 27-s + 8·29-s − 4·31-s + 3·33-s − 37-s + 39-s − 3·41-s − 4·43-s − 10·47-s + 2·49-s + 3·51-s + 9·53-s − 4·59-s + 9·61-s + 3·63-s + 4·67-s − 3·69-s − 7·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.654·21-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.522·33-s − 0.164·37-s + 0.160·39-s − 0.468·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s + 0.420·51-s + 1.23·53-s − 0.520·59-s + 1.15·61-s + 0.377·63-s + 0.488·67-s − 0.361·69-s − 0.830·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−16.24661725823891, −15.80094035897652, −15.07365327527106, −14.78411427458426, −14.07139196375910, −13.36952070961480, −13.03998019603514, −12.24058884816358, −11.75018810275777, −11.22934032196111, −10.64669540540418, −10.28994595963119, −9.494194336272716, −8.747935040678532, −8.158289171580181, −7.712616216061724, −6.857043788749007, −6.480493549032704, −5.407524097492107, −5.064137884764465, −4.593258399334252, −3.698988262564904, −2.712686293253548, −1.996656211672393, −1.110721879550951, 0,
1.110721879550951, 1.996656211672393, 2.712686293253548, 3.698988262564904, 4.593258399334252, 5.064137884764465, 5.407524097492107, 6.480493549032704, 6.857043788749007, 7.712616216061724, 8.158289171580181, 8.747935040678532, 9.494194336272716, 10.28994595963119, 10.64669540540418, 11.22934032196111, 11.75018810275777, 12.24058884816358, 13.03998019603514, 13.36952070961480, 14.07139196375910, 14.78411427458426, 15.07365327527106, 15.80094035897652, 16.24661725823891
