| L(s) = 1 | + 2.28·2-s − 6.58·3-s − 2.76·4-s + 7.90·5-s − 15.0·6-s − 15.8·7-s − 24.6·8-s + 16.3·9-s + 18.0·10-s − 26.9·11-s + 18.1·12-s + 26.9·13-s − 36.2·14-s − 52.0·15-s − 34.2·16-s − 16.0·17-s + 37.3·18-s − 34.7·19-s − 21.8·20-s + 104.·21-s − 61.6·22-s − 134.·23-s + 162.·24-s − 62.5·25-s + 61.7·26-s + 70.3·27-s + 43.7·28-s + ⋯ |
| L(s) = 1 | + 0.809·2-s − 1.26·3-s − 0.345·4-s + 0.706·5-s − 1.02·6-s − 0.854·7-s − 1.08·8-s + 0.604·9-s + 0.571·10-s − 0.737·11-s + 0.437·12-s + 0.576·13-s − 0.691·14-s − 0.895·15-s − 0.535·16-s − 0.228·17-s + 0.488·18-s − 0.419·19-s − 0.244·20-s + 1.08·21-s − 0.597·22-s − 1.22·23-s + 1.37·24-s − 0.500·25-s + 0.466·26-s + 0.501·27-s + 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3632978217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3632978217\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.28T + 8T^{2} \) |
| 3 | \( 1 + 6.58T + 27T^{2} \) |
| 5 | \( 1 - 7.90T + 125T^{2} \) |
| 7 | \( 1 + 15.8T + 343T^{2} \) |
| 11 | \( 1 + 26.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 413.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 459.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 124.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 498.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 752.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 529.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 578.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 991.T + 9.12e5T^{2} \) |
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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
−8.983670530848724749139339719736, −8.144971892753428213065428947742, −6.74843875142829921845725373480, −6.23796837392907260909743215384, −5.58241399574520570310119998964, −5.10963721230736655809957947412, −4.04268364655901892696999977426, −3.19146531855519317696893698828, −1.90698872462531977412572485183, −0.25361363432310314282099954402,
0.25361363432310314282099954402, 1.90698872462531977412572485183, 3.19146531855519317696893698828, 4.04268364655901892696999977426, 5.10963721230736655809957947412, 5.58241399574520570310119998964, 6.23796837392907260909743215384, 6.74843875142829921845725373480, 8.144971892753428213065428947742, 8.983670530848724749139339719736
