| L(s) = 1 | − 2·7-s − 6·11-s − 5·13-s + 3·17-s + 2·19-s − 6·23-s + 3·29-s − 4·31-s − 5·37-s − 6·41-s + 10·43-s − 3·49-s + 6·53-s − 12·59-s + 5·61-s − 2·67-s + 6·71-s + 73-s + 12·77-s − 10·79-s − 3·89-s + 10·91-s + 10·97-s + 6·101-s + 16·103-s − 12·107-s − 7·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.80·11-s − 1.38·13-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 0.557·29-s − 0.718·31-s − 0.821·37-s − 0.937·41-s + 1.52·43-s − 3/7·49-s + 0.824·53-s − 1.56·59-s + 0.640·61-s − 0.244·67-s + 0.712·71-s + 0.117·73-s + 1.36·77-s − 1.12·79-s − 0.317·89-s + 1.04·91-s + 1.01·97-s + 0.597·101-s + 1.57·103-s − 1.16·107-s − 0.670·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7397701982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7397701982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.65433721465009345576817835401, −7.36430582718950100855247303424, −6.45181020343502064976272020364, −5.56262751309671077277823454198, −5.21753910040511948339749772509, −4.34017919287655960879938652188, −3.31514877969543288632985631509, −2.73507607707704950623956463611, −1.95218662726383457021540188251, −0.39596173074095167700089835755,
0.39596173074095167700089835755, 1.95218662726383457021540188251, 2.73507607707704950623956463611, 3.31514877969543288632985631509, 4.34017919287655960879938652188, 5.21753910040511948339749772509, 5.56262751309671077277823454198, 6.45181020343502064976272020364, 7.36430582718950100855247303424, 7.65433721465009345576817835401
