L(s) = 1 | + (0.887 − 1.48i)3-s + (0.444 − 2.51i)5-s + (−0.612 + 0.513i)7-s + (−1.42 − 2.63i)9-s + (0.313 + 1.77i)11-s + (−2.44 + 0.888i)13-s + (−3.35 − 2.89i)15-s + (3.13 − 5.43i)17-s + (2.88 + 5.00i)19-s + (0.220 + 1.36i)21-s + (1.80 + 1.51i)23-s + (−1.45 − 0.528i)25-s + (−5.19 − 0.222i)27-s + (7.05 + 2.56i)29-s + (−4.55 − 3.81i)31-s + ⋯ |
L(s) = 1 | + (0.512 − 0.858i)3-s + (0.198 − 1.12i)5-s + (−0.231 + 0.194i)7-s + (−0.475 − 0.879i)9-s + (0.0946 + 0.536i)11-s + (−0.677 + 0.246i)13-s + (−0.865 − 0.747i)15-s + (0.760 − 1.31i)17-s + (0.662 + 1.14i)19-s + (0.0481 + 0.298i)21-s + (0.377 + 0.316i)23-s + (−0.290 − 0.105i)25-s + (−0.999 − 0.0428i)27-s + (1.31 + 0.476i)29-s + (−0.817 − 0.685i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06981 - 0.883813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06981 - 0.883813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.887 + 1.48i)T \) |
good | 5 | \( 1 + (-0.444 + 2.51i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.612 - 0.513i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.313 - 1.77i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.44 - 0.888i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.88 - 5.00i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.80 - 1.51i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.05 - 2.56i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.55 + 3.81i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.0710 + 0.123i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.49 + 2.72i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.02 - 11.4i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.93 - 2.46i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + (-0.688 + 3.90i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (10.0 - 8.39i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-13.7 + 5.01i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.27 - 5.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.483 + 0.837i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.693 + 0.252i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (6.18 + 2.25i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.90 + 5.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.957 + 5.43i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.49744190982008825775902638632, −11.51694051118419793315550475132, −9.643836977480081701438378488237, −9.298021984218293901999320074248, −8.017414852569409330142461121665, −7.25341182789921681052650926873, −5.88350235014654639357017293364, −4.72228780406298944038184190163, −2.95129293604983968274156224611, −1.31482786748747741583014245031,
2.68915013069011854075485551360, 3.59748217274914855258526080034, 5.09238858840428866563753483734, 6.40630350232032733028628958216, 7.56082134884253405957298678392, 8.686444592338594402993368080838, 9.801693942360431262807334026045, 10.51490275595068355700731457936, 11.19253327470237067421180127766, 12.57504483689732361031716747067