| L(s) = 1 | + 2-s − 0.367·3-s + 4-s + 2.32·5-s − 0.367·6-s + 1.76·7-s + 8-s − 2.86·9-s + 2.32·10-s − 4.04·11-s − 0.367·12-s + 6.40·13-s + 1.76·14-s − 0.855·15-s + 16-s + 7.48·17-s − 2.86·18-s + 4.29·19-s + 2.32·20-s − 0.650·21-s − 4.04·22-s + 2.11·23-s − 0.367·24-s + 0.412·25-s + 6.40·26-s + 2.15·27-s + 1.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.212·3-s + 0.5·4-s + 1.04·5-s − 0.150·6-s + 0.667·7-s + 0.353·8-s − 0.954·9-s + 0.735·10-s − 1.21·11-s − 0.106·12-s + 1.77·13-s + 0.472·14-s − 0.220·15-s + 0.250·16-s + 1.81·17-s − 0.675·18-s + 0.985·19-s + 0.520·20-s − 0.141·21-s − 0.861·22-s + 0.441·23-s − 0.0750·24-s + 0.0824·25-s + 1.25·26-s + 0.415·27-s + 0.333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.856496114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.856496114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 0.367T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 5.71T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 - 9.38T + 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 7.93T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 + 5.40T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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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
−8.230562316604732445348086133705, −7.83746406635900287419230187238, −6.76375672408522854347987409657, −5.87312117080745680788994760370, −5.36045858818141956807843409434, −5.21269360269175979480686797084, −3.64711884499331140944187772012, −3.11346815945125574832369132080, −2.03072914132459392842140987208, −1.10370331577301415061684358772,
1.10370331577301415061684358772, 2.03072914132459392842140987208, 3.11346815945125574832369132080, 3.64711884499331140944187772012, 5.21269360269175979480686797084, 5.36045858818141956807843409434, 5.87312117080745680788994760370, 6.76375672408522854347987409657, 7.83746406635900287419230187238, 8.230562316604732445348086133705
