L(s) = 1 | + (−1.12 + 0.649i)2-s + (−1.28 − 5.03i)3-s + (−3.15 + 5.46i)4-s + (4.00 + 2.31i)5-s + (4.71 + 4.83i)6-s + (7.19 − 4.15i)7-s − 18.5i·8-s + (−23.7 + 12.9i)9-s − 6.01·10-s + (32.0 − 18.5i)11-s + (31.5 + 8.87i)12-s + (−29.3 − 36.5i)13-s + (−5.39 + 9.35i)14-s + (6.51 − 23.1i)15-s + (−13.1 − 22.7i)16-s − 98.9·17-s + ⋯ |
L(s) = 1 | + (−0.397 + 0.229i)2-s + (−0.246 − 0.969i)3-s + (−0.394 + 0.683i)4-s + (0.358 + 0.207i)5-s + (0.320 + 0.328i)6-s + (0.388 − 0.224i)7-s − 0.821i·8-s + (−0.878 + 0.478i)9-s − 0.190·10-s + (0.879 − 0.507i)11-s + (0.759 + 0.213i)12-s + (−0.626 − 0.779i)13-s + (−0.103 + 0.178i)14-s + (0.112 − 0.398i)15-s + (−0.205 − 0.356i)16-s − 1.41·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.512673 - 0.640506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.512673 - 0.640506i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 + 5.03i)T \) |
| 13 | \( 1 + (29.3 + 36.5i)T \) |
good | 2 | \( 1 + (1.12 - 0.649i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-4.00 - 2.31i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-7.19 + 4.15i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-32.0 + 18.5i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + 98.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 138. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-35.5 + 61.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.03i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-109. - 63.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 256. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-390. - 225. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (202. + 351. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (498. - 287. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-177. - 102. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-171. - 297. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (451. + 260. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 294. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 539. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-234. - 406. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-299. + 173. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.09e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-482. + 278. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.91008667824260735005345816890, −11.79402365917696337478426651756, −10.81974910737725474607213866915, −9.216193386701832400374863304987, −8.359022827574142830265599132901, −7.22960243351084345912853212357, −6.38541797672357761993813445659, −4.63352842330652065191379632316, −2.66134860320478708184315275504, −0.52172006425934288548197066794,
1.79522841028214244102416612174, 4.19512824532124730431450442121, 5.18819244409227129208346535572, 6.40487727956605216829735748214, 8.432723990413200880639305291709, 9.459200181558258722928864588363, 9.887732177665733089398764490632, 11.18048908936502595087444265771, 11.89140029642632335110963051978, 13.54722783258645350193968886859