| L(s) = 1 | + (−0.324 + 1.83i)2-s + (0.743 + 1.56i)3-s + (−1.39 − 0.507i)4-s + (1.37 + 0.500i)5-s + (−3.11 + 0.860i)6-s + (−2.42 − 1.06i)7-s + (−0.481 + 0.833i)8-s + (−1.89 + 2.32i)9-s + (−1.36 + 2.36i)10-s + (4.29 − 1.56i)11-s + (−0.243 − 2.55i)12-s + (0.925 + 0.336i)13-s + (2.73 − 4.11i)14-s + (0.240 + 2.52i)15-s + (−3.65 − 3.06i)16-s + (0.620 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.229 + 1.29i)2-s + (0.429 + 0.903i)3-s + (−0.697 − 0.253i)4-s + (0.614 + 0.223i)5-s + (−1.27 + 0.351i)6-s + (−0.916 − 0.400i)7-s + (−0.170 + 0.294i)8-s + (−0.631 + 0.775i)9-s + (−0.431 + 0.747i)10-s + (1.29 − 0.470i)11-s + (−0.0703 − 0.738i)12-s + (0.256 + 0.0934i)13-s + (0.731 − 1.09i)14-s + (0.0619 + 0.651i)15-s + (−0.912 − 0.765i)16-s + (0.150 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287787 + 1.19754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287787 + 1.19754i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.743 - 1.56i)T \) |
| 7 | \( 1 + (2.42 + 1.06i)T \) |
| good | 2 | \( 1 + (0.324 - 1.83i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.37 - 0.500i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-4.29 + 1.56i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.925 - 0.336i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.620 + 1.07i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.718 + 1.24i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.165 - 0.939i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.18 + 1.88i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-6.28 - 2.28i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 + (-11.1 - 4.06i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.559 + 3.17i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (10.6 - 3.87i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.32 - 4.02i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.83 - 4.05i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.93 - 1.79i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.28 + 12.9i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.80 + 10.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 + (-2.45 + 13.9i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (7.82 - 2.84i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.307 - 0.532i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.991 - 5.62i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−13.72102506551260581355749038150, −11.93494056195088394504997810313, −10.71534979748085195635523020935, −9.599230147980641670500040189027, −9.048776079280709871907112044300, −7.931259630402854638937171609842, −6.56948505876294565341410335677, −6.00214177891651617823703166597, −4.46420033689462124059004815469, −2.99937067004199675213848782258,
1.34118974774337482904828733258, 2.58261974146125850629245356732, 3.81270999100914473921519899826, 6.06824812388152037640727398065, 6.80111661468236647867238129756, 8.516693396443237221447815143372, 9.391863910685965135151081716186, 9.995155751354349289940540137876, 11.42676566856940048833084060303, 12.25964182570559626977871170695
