| L(s) = 1 | + 0.466·2-s − 2.80·3-s − 1.78·4-s − 2.27·5-s − 1.31·6-s − 1.76·8-s + 4.87·9-s − 1.06·10-s + 0.320·11-s + 5.00·12-s − 0.931·13-s + 6.37·15-s + 2.74·16-s − 1.87·17-s + 2.27·18-s + 2.54·19-s + 4.04·20-s + 0.149·22-s + 6.69·23-s + 4.95·24-s + 0.159·25-s − 0.434·26-s − 5.27·27-s + 5.03·29-s + 2.97·30-s + 0.853·31-s + 4.80·32-s + ⋯ |
| L(s) = 1 | + 0.330·2-s − 1.62·3-s − 0.891·4-s − 1.01·5-s − 0.534·6-s − 0.624·8-s + 1.62·9-s − 0.335·10-s + 0.0965·11-s + 1.44·12-s − 0.258·13-s + 1.64·15-s + 0.685·16-s − 0.455·17-s + 0.536·18-s + 0.584·19-s + 0.905·20-s + 0.0318·22-s + 1.39·23-s + 1.01·24-s + 0.0319·25-s − 0.0852·26-s − 1.01·27-s + 0.934·29-s + 0.543·30-s + 0.153·31-s + 0.850·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.466T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 11 | \( 1 - 0.320T + 11T^{2} \) |
| 13 | \( 1 + 0.931T + 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 0.853T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 43 | \( 1 + 4.71T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 0.803T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 8.32T + 71T^{2} \) |
| 73 | \( 1 + 5.02T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 0.612T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−8.771082648761173060758115205108, −7.897714819210147896786155426515, −7.01028688717241056935965758118, −6.29009896749083068357641522605, −5.31945049089152810850415016431, −4.80480212951559940690788233219, −4.13822045567391175895646766983, −3.11865868635152085963125378228, −1.02366114080757446739161001452, 0,
1.02366114080757446739161001452, 3.11865868635152085963125378228, 4.13822045567391175895646766983, 4.80480212951559940690788233219, 5.31945049089152810850415016431, 6.29009896749083068357641522605, 7.01028688717241056935965758118, 7.897714819210147896786155426515, 8.771082648761173060758115205108
