| L(s) = 1 | + (0.776 − 2.89i)3-s + (2.68 + 3.20i)5-s + (4.88 − 1.77i)7-s + (−7.79 − 4.49i)9-s + (4.52 − 5.38i)11-s + (3.33 + 18.8i)13-s + (11.3 − 5.30i)15-s + (20.3 − 11.7i)17-s + (11.7 − 20.4i)19-s + (−1.36 − 15.5i)21-s + (−8.16 + 22.4i)23-s + (1.30 − 7.41i)25-s + (−19.0 + 19.0i)27-s + (21.2 + 3.74i)29-s + (−24.0 − 8.73i)31-s + ⋯ |
| L(s) = 1 | + (0.258 − 0.965i)3-s + (0.537 + 0.640i)5-s + (0.698 − 0.254i)7-s + (−0.866 − 0.499i)9-s + (0.411 − 0.489i)11-s + (0.256 + 1.45i)13-s + (0.757 − 0.353i)15-s + (1.19 − 0.692i)17-s + (0.620 − 1.07i)19-s + (−0.0648 − 0.739i)21-s + (−0.355 + 0.975i)23-s + (0.0523 − 0.296i)25-s + (−0.706 + 0.707i)27-s + (0.731 + 0.128i)29-s + (−0.774 − 0.281i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.08311 - 0.934309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08311 - 0.934309i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.776 + 2.89i)T \) |
| good | 5 | \( 1 + (-2.68 - 3.20i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.88 + 1.77i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (-4.52 + 5.38i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-3.33 - 18.8i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-20.3 + 11.7i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-11.7 + 20.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (8.16 - 22.4i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-21.2 - 3.74i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (24.0 + 8.73i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (6.81 + 11.8i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-50.7 + 8.94i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (3.55 + 2.98i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (1.64 + 4.51i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + 67.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (55.7 + 66.4i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-50.3 + 18.3i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (3.49 + 19.8i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-85.2 + 49.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (69.6 - 120. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (18.3 - 103. i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-79.2 - 13.9i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (58.0 + 33.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (112. + 94.0i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−11.25002088487119603892110384535, −9.744247701030703255454848667647, −9.018661215996072971049427724445, −7.898933977056586981088350227326, −7.07777966554097501046556063163, −6.33088594714210814959551588662, −5.21313119585942640994717694885, −3.61592450047040251095205119460, −2.35275901359460522323160275903, −1.15455580928865603422387110145,
1.42568799307406018219318056373, 3.05107150536325762757901506597, 4.25593514284406003778245102677, 5.35020968787472415353058677465, 5.86872118155642535205925210497, 7.78270875365618874859344491924, 8.366098204800790718922939683449, 9.337499498330082746299688128783, 10.16487292461335851506655685705, 10.74699888294441849448845258803
