| L(s) = 1 | + 2.03i·3-s + (0.240 + 2.22i)5-s − 1.15·9-s + 1.04·11-s − 1.15i·13-s + (−4.53 + 0.489i)15-s + 6.81i·17-s − 4.75·19-s − 3.19i·23-s + (−4.88 + 1.06i)25-s + 3.76i·27-s − 5.14·29-s + 5.57·31-s + 2.13i·33-s + 4.91i·37-s + ⋯ |
| L(s) = 1 | + 1.17i·3-s + (0.107 + 0.994i)5-s − 0.384·9-s + 0.316·11-s − 0.320i·13-s + (−1.17 + 0.126i)15-s + 1.65i·17-s − 1.08·19-s − 0.665i·23-s + (−0.976 + 0.213i)25-s + 0.723i·27-s − 0.954·29-s + 1.00·31-s + 0.371i·33-s + 0.808i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286932804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286932804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.240 - 2.22i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.03iT - 3T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 + 1.15iT - 13T^{2} \) |
| 17 | \( 1 - 6.81iT - 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + 3.19iT - 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 - 4.91iT - 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 - 9.29iT - 47T^{2} \) |
| 53 | \( 1 - 5.60iT - 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 - 5.45iT - 67T^{2} \) |
| 71 | \( 1 + 4.10T + 71T^{2} \) |
| 73 | \( 1 + 6.46iT - 73T^{2} \) |
| 79 | \( 1 + 0.612T + 79T^{2} \) |
| 83 | \( 1 + 0.275iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 7.59iT - 97T^{2} \) |
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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
−9.926447389871035250973920191257, −8.794259540368967948638771653954, −8.250223838588705344286731794731, −7.11493127388759821156250446820, −6.35228674760560155072673901824, −5.64336199928028099096677992223, −4.46149409262834740124835765936, −3.88524881510270615048053298669, −3.03971290989643209940459463783, −1.83627001335312743424414301896,
0.45372143122764870318812979755, 1.55470023195387837986119827586, 2.39996028529028800933226662873, 3.83649195145015323122068862993, 4.80488428130252058772566479854, 5.59795131586924079668982643712, 6.60595757305193818548693433303, 7.13299099779145770000277527550, 8.009722812976817185939881090323, 8.647176632055875495172168870039
