| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s + 6·13-s − 16-s + 6·17-s − 19-s − 2·20-s − 4·23-s − 25-s − 6·26-s − 2·29-s + 8·31-s − 5·32-s − 6·34-s − 10·37-s + 38-s + 6·40-s + 2·41-s − 4·43-s + 4·46-s − 12·47-s − 7·49-s + 50-s − 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s + 1.66·13-s − 1/4·16-s + 1.45·17-s − 0.229·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.371·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s − 1.64·37-s + 0.162·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.589·46-s − 1.75·47-s − 49-s + 0.141·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8683613404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8683613404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.98289657399495306851456072162, −11.63017134596575611070760390300, −10.33061644135141849320149848866, −9.849882478281281439722427012273, −8.700339636508216566718463872719, −7.962026255736907116683171399764, −6.37820814688737049661034336254, −5.30285359031178015236684552035, −3.69904169102384607719527879903, −1.48488050469889056862198933364,
1.48488050469889056862198933364, 3.69904169102384607719527879903, 5.30285359031178015236684552035, 6.37820814688737049661034336254, 7.962026255736907116683171399764, 8.700339636508216566718463872719, 9.849882478281281439722427012273, 10.33061644135141849320149848866, 11.63017134596575611070760390300, 12.98289657399495306851456072162
