| L(s) = 1 | + (−0.173 − 0.0562i)2-s + (4.20 − 3.05i)3-s + (−12.9 − 9.38i)4-s + (−6.40 − 19.7i)5-s + (−0.900 + 0.292i)6-s + (12.6 − 17.4i)7-s + (3.42 + 4.71i)8-s + (8.34 − 25.6i)9-s + 3.77i·10-s + (−24.1 − 118. i)11-s − 82.9·12-s + (121. + 39.4i)13-s + (−3.17 + 2.30i)14-s + (−87.1 − 63.3i)15-s + (78.6 + 241. i)16-s + (−88.1 + 28.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.0433 − 0.0140i)2-s + (0.467 − 0.339i)3-s + (−0.807 − 0.586i)4-s + (−0.256 − 0.788i)5-s + (−0.0250 + 0.00812i)6-s + (0.258 − 0.356i)7-s + (0.0534 + 0.0736i)8-s + (0.103 − 0.317i)9-s + 0.0377i·10-s + (−0.199 − 0.979i)11-s − 0.576·12-s + (0.718 + 0.233i)13-s + (−0.0162 + 0.0117i)14-s + (−0.387 − 0.281i)15-s + (0.307 + 0.945i)16-s + (−0.304 + 0.0990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.820567 - 0.939874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.820567 - 0.939874i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.20 + 3.05i)T \) |
| 11 | \( 1 + (24.1 + 118. i)T \) |
| good | 2 | \( 1 + (0.173 + 0.0562i)T + (12.9 + 9.40i)T^{2} \) |
| 5 | \( 1 + (6.40 + 19.7i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (-12.6 + 17.4i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-121. - 39.4i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (88.1 - 28.6i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-348. - 480. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + 279.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-682. + 939. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-11.7 + 36.0i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-375. - 272. i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (532. + 733. i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + 2.63e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.88e3 + 1.36e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (1.35e3 - 4.16e3i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-4.46e3 - 3.24e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (3.46e3 - 1.12e3i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 3.72e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (618. + 1.90e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (493. - 679. i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-7.87e3 - 2.55e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (4.44e3 - 1.44e3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 + 1.47e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (4.28e3 - 1.31e4i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−15.61109217808903552945452988603, −14.03872924820412650134279376660, −13.56155073517203404183152379419, −12.09238382754467867330135557401, −10.41865390690701254335572126540, −8.935238355185119885899097194854, −8.056437930349475826998123025494, −5.81053845734365637170573489142, −4.07149351633797903153025707741, −0.997797181300797095023909249202,
3.09419932701141953560225634490, 4.77306170548250197681276572340, 7.21548564211091467058643732388, 8.538293314400926596456533618377, 9.763154846071749221900501732513, 11.27497499010048238070225062572, 12.77996157455662464899564008960, 13.95850303294800187315633172617, 15.04061437050492058839974029141, 16.09741888753365234640570078474
