| L(s) = 1 | − 3-s + (−1 − 2i)5-s − 2i·7-s + 9-s − 2i·11-s − 2·13-s + (1 + 2i)15-s − 6i·17-s − 8i·19-s + 2i·21-s + 4i·23-s + (−3 + 4i)25-s − 27-s − 8i·29-s + 2i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.447 − 0.894i)5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s − 0.554·13-s + (0.258 + 0.516i)15-s − 1.45i·17-s − 1.83i·19-s + 0.436i·21-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s − 1.48i·29-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8238617689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8238617689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.934118290087340528483643131443, −7.36755316879797070452781670454, −6.72796400532654923666995763302, −5.71531446150005638769356676506, −4.90353287781904129128496325244, −4.52287740945337835369967547220, −3.52486949339636624436067475171, −2.43855041718298952145646574130, −0.915494956748907366418585187790, −0.32908894437133172800253090557,
1.58670216669201233258972177573, 2.49586402847518774850270032372, 3.56232073015672390161557220473, 4.27085294422757638879633628040, 5.26283856781323370648246448306, 6.03727900008917443602226944378, 6.57466678806329785547386722874, 7.38839189272073119099081381015, 8.102577254143068176026485834363, 8.740322834163485022205936096313
