L(s) = 1 | + 1.41i·2-s + 1.73·5-s + (1 − 2.44i)7-s + 2.82i·8-s + 2.44i·10-s + 1.41i·11-s − 2.44i·13-s + (3.46 + 1.41i)14-s − 4.00·16-s − 5.19·17-s + 7.34i·19-s − 2.00·22-s − 2.82i·23-s − 2.00·25-s + 3.46·26-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + 0.774·5-s + (0.377 − 0.925i)7-s + 0.999i·8-s + 0.774i·10-s + 0.426i·11-s − 0.679i·13-s + (0.925 + 0.377i)14-s − 1.00·16-s − 1.26·17-s + 1.68i·19-s − 0.426·22-s − 0.589i·23-s − 0.400·25-s + 0.679·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18979 + 0.799391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18979 + 0.799391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 2 | \( 1 - 1.41iT - 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 7.34iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 2.44iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 - 2.44iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−13.07651807571313266254565758175, −11.72216212277246911552028063491, −10.63960351580536385608052146590, −9.827499438068967724494874974637, −8.362479658156968219551841550469, −7.57985669344396046469323545879, −6.49192134645838242014543461484, −5.62524808166462633392988730764, −4.27294115020486697734705743889, −2.11618973299712712027780402745,
1.83112991167314617853490303370, 2.87821755845660599453994694347, 4.63727216477519667024161427335, 6.02741144301485468951493417722, 7.07752398678913708314848478297, 8.909766763618312675979689306802, 9.341091374372791732670732620300, 10.72731998292512551681967476542, 11.32445565682391576879601603432, 12.20003588939951673772072945868