| L(s) = 1 | − 0.667·2-s − 1.55·4-s − 3.64·7-s + 2.37·8-s + 1.15·11-s − 5.71·13-s + 2.43·14-s + 1.52·16-s − 1.37·17-s + 2.28·19-s − 0.772·22-s + 5.17·23-s + 3.81·26-s + 5.66·28-s + 5.51·29-s − 31-s − 5.76·32-s + 0.920·34-s − 2.77·37-s − 1.52·38-s + 0.122·41-s − 1.11·43-s − 1.79·44-s − 3.45·46-s − 8.92·47-s + 6.26·49-s + 8.87·52-s + ⋯ |
| L(s) = 1 | − 0.471·2-s − 0.777·4-s − 1.37·7-s + 0.838·8-s + 0.348·11-s − 1.58·13-s + 0.649·14-s + 0.381·16-s − 0.334·17-s + 0.524·19-s − 0.164·22-s + 1.07·23-s + 0.747·26-s + 1.07·28-s + 1.02·29-s − 0.179·31-s − 1.01·32-s + 0.157·34-s − 0.455·37-s − 0.247·38-s + 0.0191·41-s − 0.170·43-s − 0.271·44-s − 0.508·46-s − 1.30·47-s + 0.895·49-s + 1.23·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.667T + 2T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 37 | \( 1 + 2.77T + 37T^{2} \) |
| 41 | \( 1 - 0.122T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 - 3.42T + 53T^{2} \) |
| 59 | \( 1 + 0.542T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 + 9.97T + 97T^{2} \) |
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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
−7.61928343499209853517946794261, −6.89620268634860667305496507674, −6.46727678361990805800722254320, −5.24189748517848728867955232724, −4.91378716367899632504425375818, −3.89552213584265950912116716379, −3.19519293854290076263789412747, −2.31533616848062044190558672230, −0.943433535507476676115338031724, 0,
0.943433535507476676115338031724, 2.31533616848062044190558672230, 3.19519293854290076263789412747, 3.89552213584265950912116716379, 4.91378716367899632504425375818, 5.24189748517848728867955232724, 6.46727678361990805800722254320, 6.89620268634860667305496507674, 7.61928343499209853517946794261
