| L(s) = 1 | + 7-s + 4·11-s + 6·13-s + 23-s − 5·25-s − 2·29-s − 2·31-s + 2·37-s + 6·41-s − 4·43-s − 2·47-s + 49-s − 14·53-s + 14·59-s + 12·61-s + 4·67-s − 2·73-s + 4·77-s + 8·79-s − 4·83-s + 16·89-s + 6·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.208·23-s − 25-s − 0.371·29-s − 0.359·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s − 1.92·53-s + 1.82·59-s + 1.53·61-s + 0.488·67-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s + 1.69·89-s + 0.628·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.028799081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.028799081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.60772730125137, −14.69981642271407, −14.52096966319548, −13.89351798563925, −13.22656607166477, −12.97259422146999, −12.06522584660545, −11.59526412073058, −11.15396036273028, −10.73576679387638, −9.839581581863609, −9.389976479466560, −8.801720357130209, −8.249360624274521, −7.750258374657159, −6.882455088720502, −6.408954664335357, −5.846020136451522, −5.208849555957352, −4.293872481114227, −3.811791673559733, −3.302388763334331, −2.152872661607841, −1.496553434265379, −0.7521768959027018,
0.7521768959027018, 1.496553434265379, 2.152872661607841, 3.302388763334331, 3.811791673559733, 4.293872481114227, 5.208849555957352, 5.846020136451522, 6.408954664335357, 6.882455088720502, 7.750258374657159, 8.249360624274521, 8.801720357130209, 9.389976479466560, 9.839581581863609, 10.73576679387638, 11.15396036273028, 11.59526412073058, 12.06522584660545, 12.97259422146999, 13.22656607166477, 13.89351798563925, 14.52096966319548, 14.69981642271407, 15.60772730125137
