| L(s) = 1 | + 7-s + 6·11-s + 6·13-s + 2·17-s + 7·19-s + 8·23-s + 6·29-s − 9·31-s + 3·37-s − 10·41-s − 43-s − 2·47-s − 6·49-s + 2·53-s + 12·59-s + 3·61-s − 4·67-s − 12·71-s − 11·73-s + 6·77-s − 11·79-s − 6·83-s − 8·89-s + 6·91-s − 7·97-s − 10·101-s − 5·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s + 1.66·13-s + 0.485·17-s + 1.60·19-s + 1.66·23-s + 1.11·29-s − 1.61·31-s + 0.493·37-s − 1.56·41-s − 0.152·43-s − 0.291·47-s − 6/7·49-s + 0.274·53-s + 1.56·59-s + 0.384·61-s − 0.488·67-s − 1.42·71-s − 1.28·73-s + 0.683·77-s − 1.23·79-s − 0.658·83-s − 0.847·89-s + 0.628·91-s − 0.710·97-s − 0.995·101-s − 0.492·103-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
| \(L(1)\) |
\(\approx\) |
\(3.001663135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001663135\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.436074739433431013505368710842, −7.24837796190820374085596485876, −6.87792088292611747872418991085, −5.99752367120771169575780009951, −5.35831153338441715812635383405, −4.44601963926788065451037962197, −3.57630690669959709644476804142, −3.11968911696280829462177886716, −1.43513918144507314701996451303, −1.16988442089926654778950218056,
1.16988442089926654778950218056, 1.43513918144507314701996451303, 3.11968911696280829462177886716, 3.57630690669959709644476804142, 4.44601963926788065451037962197, 5.35831153338441715812635383405, 5.99752367120771169575780009951, 6.87792088292611747872418991085, 7.24837796190820374085596485876, 8.436074739433431013505368710842
