| L(s) = 1 | − 1.94·2-s + 2.63·3-s + 1.79·4-s + 4.26·5-s − 5.12·6-s − 7-s + 0.397·8-s + 3.92·9-s − 8.30·10-s + 3.75·11-s + 4.72·12-s + 13-s + 1.94·14-s + 11.2·15-s − 4.36·16-s − 17-s − 7.65·18-s − 2.80·19-s + 7.65·20-s − 2.63·21-s − 7.31·22-s + 4.26·23-s + 1.04·24-s + 13.1·25-s − 1.94·26-s + 2.44·27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 1.51·3-s + 0.897·4-s + 1.90·5-s − 2.09·6-s − 0.377·7-s + 0.140·8-s + 1.30·9-s − 2.62·10-s + 1.13·11-s + 1.36·12-s + 0.277·13-s + 0.520·14-s + 2.89·15-s − 1.09·16-s − 0.242·17-s − 1.80·18-s − 0.643·19-s + 1.71·20-s − 0.574·21-s − 1.55·22-s + 0.889·23-s + 0.213·24-s + 2.63·25-s − 0.382·26-s + 0.470·27-s − 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097678377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097678377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 - 0.175T + 29T^{2} \) |
| 31 | \( 1 + 8.79T + 31T^{2} \) |
| 37 | \( 1 + 0.347T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 + 2.72T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 9.87T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.218887480844692709832502680899, −8.946682552293046035310376768797, −8.345234044350859643840362716462, −7.08989932981854374393636560716, −6.66771895618846982691841717726, −5.55351532495595208401925909543, −4.15918954632013216365813481298, −2.93269997377280370839852236220, −2.01066600767262155099646381267, −1.37969703136428766167392675165,
1.37969703136428766167392675165, 2.01066600767262155099646381267, 2.93269997377280370839852236220, 4.15918954632013216365813481298, 5.55351532495595208401925909543, 6.66771895618846982691841717726, 7.08989932981854374393636560716, 8.345234044350859643840362716462, 8.946682552293046035310376768797, 9.218887480844692709832502680899
