| L(s) = 1 | − 1.61·2-s − 3-s + 0.604·4-s − 3.65·5-s + 1.61·6-s + 0.985·7-s + 2.25·8-s + 9-s + 5.90·10-s − 11-s − 0.604·12-s + 2.65·13-s − 1.59·14-s + 3.65·15-s − 4.84·16-s − 6.25·17-s − 1.61·18-s + 1.35·19-s − 2.21·20-s − 0.985·21-s + 1.61·22-s + 2.24·23-s − 2.25·24-s + 8.37·25-s − 4.28·26-s − 27-s + 0.596·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.302·4-s − 1.63·5-s + 0.658·6-s + 0.372·7-s + 0.796·8-s + 0.333·9-s + 1.86·10-s − 0.301·11-s − 0.174·12-s + 0.735·13-s − 0.425·14-s + 0.944·15-s − 1.21·16-s − 1.51·17-s − 0.380·18-s + 0.310·19-s − 0.494·20-s − 0.215·21-s + 0.344·22-s + 0.468·23-s − 0.459·24-s + 1.67·25-s − 0.839·26-s − 0.192·27-s + 0.112·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2911414628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2911414628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 - 0.985T + 7T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 6.25T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 - 0.305T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 8.53T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.866026060715549209853003117868, −8.515846038299413664185839355652, −7.57851138425862018949073357663, −7.26633142806898001021441397209, −6.22123396854334781419973302249, −4.87943611271415801719016330392, −4.37263368804407505931998696721, −3.38749889958203954860644188672, −1.73090014436218755304954187064, −0.44435693632297068627803877352,
0.44435693632297068627803877352, 1.73090014436218755304954187064, 3.38749889958203954860644188672, 4.37263368804407505931998696721, 4.87943611271415801719016330392, 6.22123396854334781419973302249, 7.26633142806898001021441397209, 7.57851138425862018949073357663, 8.515846038299413664185839355652, 8.866026060715549209853003117868
