| L(s) = 1 | + 1.28·2-s + 3·3-s − 6.36·4-s − 16.8·5-s + 3.84·6-s − 18.3·8-s + 9·9-s − 21.6·10-s + 11·11-s − 19.0·12-s + 68.0·13-s − 50.6·15-s + 27.3·16-s − 119.·17-s + 11.5·18-s − 16.8·19-s + 107.·20-s + 14.0·22-s + 199.·23-s − 55.1·24-s + 160.·25-s + 87.1·26-s + 27·27-s + 181.·29-s − 64.8·30-s + 31.5·31-s + 182.·32-s + ⋯ |
| L(s) = 1 | + 0.452·2-s + 0.577·3-s − 0.795·4-s − 1.51·5-s + 0.261·6-s − 0.812·8-s + 0.333·9-s − 0.683·10-s + 0.301·11-s − 0.459·12-s + 1.45·13-s − 0.871·15-s + 0.427·16-s − 1.69·17-s + 0.150·18-s − 0.203·19-s + 1.20·20-s + 0.136·22-s + 1.81·23-s − 0.469·24-s + 1.28·25-s + 0.657·26-s + 0.192·27-s + 1.15·29-s − 0.394·30-s + 0.182·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.28T + 8T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 13 | \( 1 - 68.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 199.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 75.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 408.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 563.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 622.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 280.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 807.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 710.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 816.T + 9.12e5T^{2} \) |
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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
−8.749751572753281887536821243690, −8.107396576828233181632676053028, −6.99681633601845302375232175781, −6.33557875082402926085183968144, −4.90560563203889395717147777637, −4.34589680548284545824929592995, −3.62170844863136877246063501101, −2.93402170983701209335113159033, −1.16391906969938763467254578748, 0,
1.16391906969938763467254578748, 2.93402170983701209335113159033, 3.62170844863136877246063501101, 4.34589680548284545824929592995, 4.90560563203889395717147777637, 6.33557875082402926085183968144, 6.99681633601845302375232175781, 8.107396576828233181632676053028, 8.749751572753281887536821243690
