| L(s) = 1 | − 2-s + 1.34·3-s + 4-s − 2.87·5-s − 1.34·6-s + 7-s − 8-s − 1.18·9-s + 2.87·10-s − 0.652·11-s + 1.34·12-s + 1.53·13-s − 14-s − 3.87·15-s + 16-s + 5.63·17-s + 1.18·18-s − 2.87·20-s + 1.34·21-s + 0.652·22-s + 0.369·23-s − 1.34·24-s + 3.29·25-s − 1.53·26-s − 5.63·27-s + 28-s + 3.77·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.777·3-s + 0.5·4-s − 1.28·5-s − 0.550·6-s + 0.377·7-s − 0.353·8-s − 0.394·9-s + 0.910·10-s − 0.196·11-s + 0.388·12-s + 0.424·13-s − 0.267·14-s − 1.00·15-s + 0.250·16-s + 1.36·17-s + 0.279·18-s − 0.643·20-s + 0.294·21-s + 0.139·22-s + 0.0770·23-s − 0.275·24-s + 0.658·25-s − 0.300·26-s − 1.08·27-s + 0.188·28-s + 0.700·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 11 | \( 1 + 0.652T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 23 | \( 1 - 0.369T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 0.403T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−8.211826329426982325368248771049, −7.38929306430245737405458067310, −6.83139639748571341907716405261, −5.67212422220794682046227082179, −4.96140584927743854577816492776, −3.63440956760380372215173011894, −3.47484667335726850016520009413, −2.39334166901707793521730267709, −1.27112120808680496365448716562, 0,
1.27112120808680496365448716562, 2.39334166901707793521730267709, 3.47484667335726850016520009413, 3.63440956760380372215173011894, 4.96140584927743854577816492776, 5.67212422220794682046227082179, 6.83139639748571341907716405261, 7.38929306430245737405458067310, 8.211826329426982325368248771049
