| L(s) = 1 | + (−1.38 − 0.283i)2-s + (0.0404 − 0.0976i)3-s + (1.83 + 0.784i)4-s + (0.453 − 0.891i)5-s + (−0.0837 + 0.123i)6-s + (−0.683 − 2.84i)7-s + (−2.32 − 1.60i)8-s + (2.11 + 2.11i)9-s + (−0.881 + 1.10i)10-s + (−2.02 + 2.37i)11-s + (0.151 − 0.147i)12-s + (2.83 + 4.63i)13-s + (0.140 + 4.13i)14-s + (−0.0686 − 0.0804i)15-s + (2.76 + 2.88i)16-s + (−0.142 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.200i)2-s + (0.0233 − 0.0563i)3-s + (0.919 + 0.392i)4-s + (0.203 − 0.398i)5-s + (−0.0341 + 0.0505i)6-s + (−0.258 − 1.07i)7-s + (−0.822 − 0.568i)8-s + (0.704 + 0.704i)9-s + (−0.278 + 0.349i)10-s + (−0.610 + 0.714i)11-s + (0.0436 − 0.0427i)12-s + (0.787 + 1.28i)13-s + (0.0375 + 1.10i)14-s + (−0.0177 − 0.0207i)15-s + (0.691 + 0.721i)16-s + (−0.0345 − 0.439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08644 - 0.0861443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08644 - 0.0861443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.283i)T \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
| 41 | \( 1 + (-4.00 - 4.99i)T \) |
| good | 3 | \( 1 + (-0.0404 + 0.0976i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (0.683 + 2.84i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (2.02 - 2.37i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.83 - 4.63i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.142 + 1.81i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-4.93 - 3.02i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.442 - 0.321i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.000106 + 0.00135i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-1.75 + 5.40i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.464 - 1.42i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.260 + 1.64i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.02 + 4.26i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 0.847i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (1.88 - 2.60i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.321 - 2.03i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-2.92 + 2.49i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (10.5 + 9.01i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-5.95 + 5.95i)T - 73iT^{2} \) |
| 79 | \( 1 + (-7.65 - 3.17i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 3.14iT - 83T^{2} \) |
| 89 | \( 1 + (1.78 - 0.427i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (4.19 - 3.58i)T + (15.1 - 95.8i)T^{2} \) |
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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
−9.995004623285708759122083688534, −9.628645297687739938027236014948, −8.562947916141521309567460936672, −7.50945163656710976214664377462, −7.22216640400447802804592791995, −6.08860951702465975291124371249, −4.69044805931083901233376907227, −3.72316311428003393804283245733, −2.17728716754767757945427632386, −1.11306811751494409466262525953,
0.956720460871638473082525534963, 2.61720917532980487884133080406, 3.38617170961839297265931782069, 5.40043825405285841521449340168, 5.94869201680782451868941801797, 6.86955927794823316813324516634, 7.82759753087619835468242699139, 8.687626978885702318114781017305, 9.299146343031975127274853280480, 10.22873046008615498555443937846
