| L(s) = 1 | + (0.220 − 0.0388i)2-s + (−0.104 − 1.72i)3-s + (−1.83 + 0.666i)4-s + (0.595 − 3.37i)5-s + (−0.0900 − 0.376i)6-s + (−2.47 + 0.925i)7-s + (−0.764 + 0.441i)8-s + (−2.97 + 0.360i)9-s − 0.766i·10-s + (−0.563 + 0.0994i)11-s + (1.34 + 3.09i)12-s + (0.663 − 0.790i)13-s + (−0.509 + 0.299i)14-s + (−5.89 − 0.677i)15-s + (2.83 − 2.38i)16-s + 3.70·17-s + ⋯ |
| L(s) = 1 | + (0.155 − 0.0274i)2-s + (−0.0602 − 0.998i)3-s + (−0.916 + 0.333i)4-s + (0.266 − 1.50i)5-s + (−0.0367 − 0.153i)6-s + (−0.936 + 0.349i)7-s + (−0.270 + 0.156i)8-s + (−0.992 + 0.120i)9-s − 0.242i·10-s + (−0.170 + 0.0299i)11-s + (0.388 + 0.894i)12-s + (0.183 − 0.219i)13-s + (−0.136 + 0.0801i)14-s + (−1.52 − 0.174i)15-s + (0.709 − 0.595i)16-s + 0.898·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.325475 - 0.750217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325475 - 0.750217i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.104 + 1.72i)T \) |
| 7 | \( 1 + (2.47 - 0.925i)T \) |
| good | 2 | \( 1 + (-0.220 + 0.0388i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.595 + 3.37i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.563 - 0.0994i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.663 + 0.790i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 8.40iT - 19T^{2} \) |
| 23 | \( 1 + (-2.38 + 2.84i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.81 - 3.34i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.645 - 1.77i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.08 - 3.61i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.371 - 0.311i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.66 + 2.06i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (5.96 + 2.17i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 6.27i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.04 - 5.07i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.51 - 4.16i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.67 - 9.49i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.452 + 0.261i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.85 + 3.95i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.65 + 9.37i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (12.7 - 10.7i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-3.46 + 9.52i)T + (-74.3 - 62.3i)T^{2} \) |
| 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
−12.50164282820932012118845364156, −11.77383998961964406965670670006, −9.943527832571527422400207855166, −8.798940866450400043190304206757, −8.542121376004328991314786980078, −7.03344958877681611177146985708, −5.63957096925720771695494808684, −4.81620813158785649150221157795, −2.99123680955830149535473346487, −0.73700504715303239722578535309,
3.18404314829351644227747336468, 3.94168307845719214046356335722, 5.60098461797768484566255997680, 6.36265291932872465023376830785, 7.912975398361024958760793927005, 9.384135431020873178998888088055, 10.11111341614009137071837113940, 10.48874046365871492042295573121, 11.81443929037893448547542829207, 13.17652509013323320337163201738
