L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.500 − 0.866i)8-s + (−1.43 − 0.524i)11-s + (−0.939 − 0.342i)16-s + (0.939 − 1.62i)17-s + (−0.173 − 0.300i)19-s + (−1.43 + 0.524i)22-s + (0.766 − 0.642i)25-s + (−0.939 + 0.342i)32-s + (−0.326 − 1.85i)34-s + (−0.326 − 0.118i)38-s + (0.266 + 0.223i)41-s + (1.76 + 0.642i)43-s + (−0.766 + 1.32i)44-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.500 − 0.866i)8-s + (−1.43 − 0.524i)11-s + (−0.939 − 0.342i)16-s + (0.939 − 1.62i)17-s + (−0.173 − 0.300i)19-s + (−1.43 + 0.524i)22-s + (0.766 − 0.642i)25-s + (−0.939 + 0.342i)32-s + (−0.326 − 1.85i)34-s + (−0.326 − 0.118i)38-s + (0.266 + 0.223i)41-s + (1.76 + 0.642i)43-s + (−0.766 + 1.32i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.542710830\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542710830\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 11 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.432871347566459421053345805051, −8.344565407491048673003847220759, −7.49272207104592012218743431377, −6.63080297655535878647514745409, −5.57893203631644462957605568821, −5.12610921671157832061077036995, −4.20744685159412530731377526403, −2.93715750199233864845552312308, −2.59575344926785343145080441703, −0.880761588254465777176600955243,
1.98329943019660303178165381658, 3.08093391364812346600425015509, 3.94533710061696132756330164745, 4.93241644630640544412886096179, 5.59574750818720292826508148695, 6.33022766157915950753174748220, 7.35065714939848186153781785102, 7.892619999945132250679425370082, 8.529161060140867547337167169628, 9.568014686279940411658052639629