L(s) = 1 | − 0.895·2-s − 1.19·4-s − 5.08·7-s + 2.86·8-s − 2.64·11-s − 2.13·13-s + 4.55·14-s − 0.167·16-s + 7.75·17-s + 3.08·19-s + 2.36·22-s − 6.14·23-s + 1.90·26-s + 6.09·28-s + 4.13·29-s − 2.74·31-s − 5.57·32-s − 6.93·34-s + 0.0157·37-s − 2.76·38-s − 3.72·41-s + 3.81·43-s + 3.16·44-s + 5.49·46-s − 0.897·47-s + 18.9·49-s + 2.55·52-s + ⋯ |
L(s) = 1 | − 0.633·2-s − 0.599·4-s − 1.92·7-s + 1.01·8-s − 0.796·11-s − 0.591·13-s + 1.21·14-s − 0.0419·16-s + 1.87·17-s + 0.708·19-s + 0.504·22-s − 1.28·23-s + 0.374·26-s + 1.15·28-s + 0.767·29-s − 0.492·31-s − 0.985·32-s − 1.19·34-s + 0.00259·37-s − 0.448·38-s − 0.582·41-s + 0.581·43-s + 0.477·44-s + 0.810·46-s − 0.130·47-s + 2.70·49-s + 0.354·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 + 0.895T + 2T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 0.0157T + 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 3.81T + 43T^{2} \) |
| 47 | \( 1 + 0.897T + 47T^{2} \) |
| 53 | \( 1 + 9.26T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 - 0.0828T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 - 0.723T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−7.81058730185909344200621613029, −7.28899537353300240536336856155, −6.41188424217162627672538402766, −5.59185803939087134455410382074, −5.04867974477250800709716166305, −3.82335772724900092353041272270, −3.34384667476243493924791062447, −2.39621231298825841864660345969, −0.928550613724677606401776795234, 0,
0.928550613724677606401776795234, 2.39621231298825841864660345969, 3.34384667476243493924791062447, 3.82335772724900092353041272270, 5.04867974477250800709716166305, 5.59185803939087134455410382074, 6.41188424217162627672538402766, 7.28899537353300240536336856155, 7.81058730185909344200621613029