L(s) = 1 | − 2i·2-s + (4.33 − 2.86i)3-s − 4·4-s + 13.9·5-s + (−5.72 − 8.67i)6-s + 0.843i·7-s + 8i·8-s + (10.5 − 24.8i)9-s − 27.9i·10-s + 54.5·11-s + (−17.3 + 11.4i)12-s − 40.5·13-s + 1.68·14-s + (60.6 − 40.0i)15-s + 16·16-s − 3.33·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.834 − 0.551i)3-s − 0.5·4-s + 1.25·5-s + (−0.389 − 0.590i)6-s + 0.0455i·7-s + 0.353i·8-s + (0.392 − 0.919i)9-s − 0.885i·10-s + 1.49·11-s + (−0.417 + 0.275i)12-s − 0.864·13-s + 0.0321·14-s + (1.04 − 0.689i)15-s + 0.250·16-s − 0.0476·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.82137 - 1.72820i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82137 - 1.72820i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + (-4.33 + 2.86i)T \) |
| 23 | \( 1 + (95.1 + 55.8i)T \) |
good | 5 | \( 1 - 13.9T + 125T^{2} \) |
| 7 | \( 1 - 0.843iT - 343T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.33T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.4iT - 6.85e3T^{2} \) |
| 29 | \( 1 - 35.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 55.4iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 98.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 457. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 256. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 102. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 650. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 56.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 266. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 321.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 773. iT - 9.12e5T^{2} \) |
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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
−12.55245595521946755512113851382, −11.68822428038467041786208343589, −10.08831464818940004567877059140, −9.461510598093305603382790865331, −8.579939367658663302781185451855, −7.05765281074115110674424255029, −5.88198121562305985879877325452, −4.07455477236119446084139325683, −2.51112370300735174398475053215, −1.43078116170214571983657135626,
1.99289051129356206012971912778, 3.82886312297803820672484186545, 5.18594509790081715708907240058, 6.42750033914413785224419967531, 7.63419462752220472828650869379, 9.081201275734081756404816870085, 9.461759509543311667132911044848, 10.47988187326097759982186861214, 12.14474281172366677336566441529, 13.48231447877081946587387376206