| L(s) = 1 | + 7-s + 2·11-s + 2·13-s + 6·19-s − 6·29-s + 10·31-s − 6·41-s + 8·43-s + 12·47-s + 49-s − 6·53-s + 6·61-s − 4·67-s − 6·71-s − 14·73-s + 2·77-s + 4·79-s + 6·89-s + 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.554·13-s + 1.37·19-s − 1.11·29-s + 1.79·31-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.768·61-s − 0.488·67-s − 0.712·71-s − 1.63·73-s + 0.227·77-s + 0.450·79-s + 0.635·89-s + 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.446546343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446546343\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.84624038180310, −13.30154784773642, −12.85896323847688, −12.07317814271242, −11.76125283485626, −11.51553278270674, −10.66364307840690, −10.48432109139342, −9.702576371327064, −9.264418416539186, −8.885669990626018, −8.179308470860034, −7.812523570079712, −7.161605874088777, −6.791371720170229, −5.936909194062895, −5.742374800635292, −5.012486992721984, −4.383256473849509, −3.930463904036851, −3.227239853376932, −2.715221402973958, −1.857962353338214, −1.232099657884534, −0.6407659503579573,
0.6407659503579573, 1.232099657884534, 1.857962353338214, 2.715221402973958, 3.227239853376932, 3.930463904036851, 4.383256473849509, 5.012486992721984, 5.742374800635292, 5.936909194062895, 6.791371720170229, 7.161605874088777, 7.812523570079712, 8.179308470860034, 8.885669990626018, 9.264418416539186, 9.702576371327064, 10.48432109139342, 10.66364307840690, 11.51553278270674, 11.76125283485626, 12.07317814271242, 12.85896323847688, 13.30154784773642, 13.84624038180310
