| L(s) = 1 | + (0.924 + 2.53i)2-s + (−0.795 − 1.53i)3-s + (−4.05 + 3.40i)4-s + (0.268 + 1.52i)5-s + (3.17 − 3.44i)6-s + (−1.64 + 2.07i)7-s + (−7.72 − 4.45i)8-s + (−1.73 + 2.44i)9-s + (−3.61 + 2.08i)10-s + (3.67 + 0.647i)11-s + (8.47 + 3.53i)12-s + (−0.202 + 0.555i)13-s + (−6.78 − 2.26i)14-s + (2.12 − 1.62i)15-s + (2.34 − 13.2i)16-s + (−2.21 − 3.83i)17-s + ⋯ |
| L(s) = 1 | + (0.653 + 1.79i)2-s + (−0.459 − 0.888i)3-s + (−2.02 + 1.70i)4-s + (0.119 + 0.680i)5-s + (1.29 − 1.40i)6-s + (−0.621 + 0.783i)7-s + (−2.72 − 1.57i)8-s + (−0.578 + 0.816i)9-s + (−1.14 + 0.659i)10-s + (1.10 + 0.195i)11-s + (2.44 + 1.02i)12-s + (−0.0560 + 0.153i)13-s + (−1.81 − 0.604i)14-s + (0.548 − 0.418i)15-s + (0.585 − 3.32i)16-s + (−0.537 − 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0956368 + 1.12167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0956368 + 1.12167i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.795 + 1.53i)T \) |
| 7 | \( 1 + (1.64 - 2.07i)T \) |
| good | 2 | \( 1 + (-0.924 - 2.53i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.268 - 1.52i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-3.67 - 0.647i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.202 - 0.555i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.21 + 3.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.88 - 3.39i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 - 1.45i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.530 + 1.45i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.97 - 4.74i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.87 + 3.24i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.26 - 0.459i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.06 - 6.03i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.70 + 3.95i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 2.96iT - 53T^{2} \) |
| 59 | \( 1 + (0.751 + 4.26i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.03 - 7.19i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.30 - 1.56i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.85 + 2.80i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.50 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.68 - 2.79i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.97 + 1.80i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.95 + 5.11i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-18.3 - 3.24i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.32600553585948655512005104182, −12.30776980580339501093213441381, −11.64762268408342437148966371628, −9.618645687512596656326579037812, −8.633945020191574397344893071036, −7.38375546164918982243556248702, −6.71384234288158187982763251215, −6.03252796058587454107688542773, −4.96747980751807316610265445509, −3.18053582291439509574177122086,
0.956011907164848546056833043654, 3.23179607687669659996402002811, 4.16211474409869128323020647058, 5.04192537141919094683625774748, 6.32352355536196973646156899908, 8.891793225583082789950728566448, 9.492393668622896190212502040774, 10.35778867394657476769525269548, 11.18395128726820349136041313185, 11.96228924662990051959644756193
