| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s − 2·9-s + 2·10-s − 5·11-s − 12-s − 2·13-s − 14-s + 2·15-s + 16-s − 3·17-s + 2·18-s − 2·19-s − 2·20-s − 21-s + 5·22-s + 4·23-s + 24-s − 25-s + 2·26-s + 5·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 1.50·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.06743516713373287162354131658, −11.08218814744580086594606755479, −10.67416575071043460588285749703, −9.168275289034099779760620307092, −8.078731312648121835321718035189, −7.36625732552646488198525538003, −5.88341646402151712569560354820, −4.65218853180790869047123049112, −2.69630329480699135407030849925, 0,
2.69630329480699135407030849925, 4.65218853180790869047123049112, 5.88341646402151712569560354820, 7.36625732552646488198525538003, 8.078731312648121835321718035189, 9.168275289034099779760620307092, 10.67416575071043460588285749703, 11.08218814744580086594606755479, 12.06743516713373287162354131658
