| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 6·11-s − 13-s + 3·21-s + 23-s − 9·27-s − 3·29-s + 3·31-s − 18·33-s + 8·37-s + 3·39-s + 9·41-s + 4·43-s + 13·47-s + 49-s − 4·53-s − 4·59-s + 2·61-s − 6·63-s − 4·67-s − 3·69-s + 5·71-s − 3·73-s − 6·77-s − 12·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 0.654·21-s + 0.208·23-s − 1.73·27-s − 0.557·29-s + 0.538·31-s − 3.13·33-s + 1.31·37-s + 0.480·39-s + 1.40·41-s + 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.549·53-s − 0.520·59-s + 0.256·61-s − 0.755·63-s − 0.488·67-s − 0.361·69-s + 0.593·71-s − 0.351·73-s − 0.683·77-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610131014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610131014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.24282093135259, −13.72123376617504, −12.92025527535809, −12.65757283186855, −12.06339272204154, −11.75624293705140, −11.25615143758669, −10.84779089543667, −10.27569682345686, −9.692003649189914, −9.210436854805638, −8.858348449133641, −7.722357008612807, −7.390754281382156, −6.706610660597112, −6.284022305632861, −5.938980935406601, −5.394412743708508, −4.533378801535727, −4.272802348339132, −3.668819410857720, −2.724791560912354, −1.824095076391545, −0.9866521422317045, −0.6193452208635697,
0.6193452208635697, 0.9866521422317045, 1.824095076391545, 2.724791560912354, 3.668819410857720, 4.272802348339132, 4.533378801535727, 5.394412743708508, 5.938980935406601, 6.284022305632861, 6.706610660597112, 7.390754281382156, 7.722357008612807, 8.858348449133641, 9.210436854805638, 9.692003649189914, 10.27569682345686, 10.84779089543667, 11.25615143758669, 11.75624293705140, 12.06339272204154, 12.65757283186855, 12.92025527535809, 13.72123376617504, 14.24282093135259
