| L(s) = 1 | + (−2.08 − 1.63i)2-s + (0.518 − 0.728i)3-s + (1.18 + 4.87i)4-s + (1.02 − 0.196i)5-s + (−2.27 + 0.667i)6-s + (2.29 − 1.31i)7-s + (3.31 − 7.26i)8-s + (0.719 + 2.07i)9-s + (−2.44 − 1.26i)10-s + (1.93 − 1.52i)11-s + (4.16 + 1.66i)12-s + (0.956 − 0.614i)13-s + (−6.93 − 1.01i)14-s + (0.386 − 0.846i)15-s + (−9.89 + 5.10i)16-s + (−4.12 − 3.93i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 − 1.15i)2-s + (0.299 − 0.420i)3-s + (0.591 + 2.43i)4-s + (0.456 − 0.0880i)5-s + (−0.928 + 0.272i)6-s + (0.867 − 0.497i)7-s + (1.17 − 2.56i)8-s + (0.239 + 0.692i)9-s + (−0.774 − 0.399i)10-s + (0.583 − 0.458i)11-s + (1.20 + 0.481i)12-s + (0.265 − 0.170i)13-s + (−1.85 − 0.271i)14-s + (0.0997 − 0.218i)15-s + (−2.47 + 1.27i)16-s + (−1.00 − 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.487760 - 0.552856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487760 - 0.552856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.29 + 1.31i)T \) |
| 23 | \( 1 + (-1.92 + 4.39i)T \) |
| good | 2 | \( 1 + (2.08 + 1.63i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.518 + 0.728i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-1.02 + 0.196i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 1.52i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.956 + 0.614i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.12 + 3.93i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.81 - 3.63i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-2.28 + 0.671i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-7.46 + 0.712i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.366 - 1.05i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (1.34 + 1.55i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.416 + 0.912i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (6.01 - 10.4i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.437 - 9.18i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (3.61 + 1.86i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (6.01 + 8.44i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-4.45 + 1.78i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.86 - 12.9i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.311 - 1.28i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.606 - 12.7i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (3.16 - 3.65i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.39 + 0.419i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-9.34 - 10.7i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.32774253795455801818750622169, −11.23741616330430501443160141844, −10.66608473668947223248330605541, −9.624627869097513771039167520724, −8.502402583996789899105220116321, −7.958087796231865113998080260638, −6.72128552240659376922092600309, −4.37441732236528260471106902331, −2.55266901787677307516931639183, −1.35941448038810649029035995767,
1.78263346070349967716894360956, 4.60974805633622181880621878382, 6.12147708767383880957799090748, 6.86198485022187384949907133817, 8.305228505057127460754123854048, 8.907584429500394322169033591316, 9.725737866925061268853440272374, 10.67131431189573667520962148503, 11.77440608618211474701718661116, 13.54323914773122974861031130628
