| L(s) = 1 | + i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 6·11-s − 2i·13-s + (1 − 2i)15-s + i·17-s + 2·21-s + 4i·23-s + (3 + 4i)25-s − i·27-s + 8·31-s − 6i·33-s + (−2 + 4i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 1.80·11-s − 0.554i·13-s + (0.258 − 0.516i)15-s + 0.242i·17-s + 0.436·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s + 1.43·31-s − 1.04i·33-s + (−0.338 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9089510069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9089510069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−9.289561154883854979375807121057, −8.247169377666781407654251839431, −7.911433378652973613321711861174, −7.18398207753420309363629556555, −5.93697951462753148183368962019, −5.04271101581489130900285070541, −4.48415631837275459601018936001, −3.50902819497367371071837608382, −2.69665413170645044314226481025, −0.873241431082976301405158368076,
0.43175973370645989404736552094, 2.35354780256471794224224286057, 2.76625839346010426031716458555, 4.05562548422323838091386482555, 5.04505667660397636341883823770, 5.84511644137409164978919191189, 6.80558640104224134977295148963, 7.46180145379143892792371058079, 8.218653123071807660987326044218, 8.669146525120559742697353251849
