| L(s) = 1 | + 2·7-s − 3·11-s + 3·13-s − 2·17-s + 4·19-s − 5·23-s + 6·31-s − 3·37-s + 2·41-s + 4·43-s + 5·47-s − 3·49-s + 4·53-s + 9·59-s + 3·61-s − 2·67-s + 9·71-s + 14·73-s − 6·77-s − 14·79-s − 14·89-s + 6·91-s + 7·97-s − 10·101-s + 14·103-s + 107-s − 10·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.904·11-s + 0.832·13-s − 0.485·17-s + 0.917·19-s − 1.04·23-s + 1.07·31-s − 0.493·37-s + 0.312·41-s + 0.609·43-s + 0.729·47-s − 3/7·49-s + 0.549·53-s + 1.17·59-s + 0.384·61-s − 0.244·67-s + 1.06·71-s + 1.63·73-s − 0.683·77-s − 1.57·79-s − 1.48·89-s + 0.628·91-s + 0.710·97-s − 0.995·101-s + 1.37·103-s + 0.0966·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.090860464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090860464\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.223202359433904359717023994075, −7.57092793822024236636964024173, −6.79833576305068902333883824322, −5.89544487179932793097913162552, −5.32986233521088524186705633343, −4.51807038989163700560438133539, −3.76273631101172723215326151957, −2.75347505823124376463949630368, −1.90675965323187635787886679021, −0.789245370604955046211651473145,
0.789245370604955046211651473145, 1.90675965323187635787886679021, 2.75347505823124376463949630368, 3.76273631101172723215326151957, 4.51807038989163700560438133539, 5.32986233521088524186705633343, 5.89544487179932793097913162552, 6.79833576305068902333883824322, 7.57092793822024236636964024173, 8.223202359433904359717023994075
