| L(s) = 1 | + (1.39 − 0.245i)2-s + (2.89 − 0.783i)3-s + (1.87 − 0.684i)4-s + (−3.55 + 4.24i)5-s + (3.84 − 1.80i)6-s + (−10.2 − 3.73i)7-s + (2.44 − 1.41i)8-s + (7.77 − 4.53i)9-s + (−3.91 + 6.77i)10-s + (2.64 + 3.15i)11-s + (4.90 − 3.45i)12-s + (−0.953 + 5.40i)13-s + (−15.2 − 2.68i)14-s + (−6.98 + 15.0i)15-s + (3.06 − 2.57i)16-s + (4.10 + 2.37i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (0.965 − 0.261i)3-s + (0.469 − 0.171i)4-s + (−0.711 + 0.848i)5-s + (0.640 − 0.300i)6-s + (−1.46 − 0.533i)7-s + (0.306 − 0.176i)8-s + (0.863 − 0.504i)9-s + (−0.391 + 0.677i)10-s + (0.240 + 0.286i)11-s + (0.408 − 0.287i)12-s + (−0.0733 + 0.416i)13-s + (−1.08 − 0.191i)14-s + (−0.465 + 1.00i)15-s + (0.191 − 0.160i)16-s + (0.241 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.78451 - 0.175909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78451 - 0.175909i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.245i)T \) |
| 3 | \( 1 + (-2.89 + 0.783i)T \) |
| good | 5 | \( 1 + (3.55 - 4.24i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (10.2 + 3.73i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-2.64 - 3.15i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.953 - 5.40i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-4.10 - 2.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (17.1 + 29.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-12.6 - 34.6i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (5.18 - 0.914i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-34.9 + 12.7i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (12.1 - 21.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (22.6 + 3.98i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-39.1 + 32.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-28.3 + 77.8i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 16.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (45.6 - 54.4i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (74.1 + 26.9i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (12.1 - 68.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (65.9 + 38.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-25.5 - 44.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-5.51 - 31.2i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-28.7 + 5.06i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (69.2 - 40.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-36.3 + 30.5i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−15.20216216224770645491969665634, −13.77568513012028554355616473304, −13.14160855340712958238437132175, −11.87304591500109595285912489799, −10.44874976614386010326234760048, −9.212999082114188670185167904760, −7.30357415650357885706083834594, −6.67814947063367080078474225480, −3.98932022230880128229667946263, −2.93446409796655742873161744395,
3.06368573362676867583507205708, 4.37797044776769704730544168683, 6.26239629548866079371251223349, 7.988111572277046639272926992061, 9.019511237889781641579460693233, 10.37587748409041762662043185921, 12.48076905814130625400107235185, 12.63355712919703711768016694882, 14.08222796410064882614072204960, 15.17871365251724711628717397418
