| L(s) = 1 | + (−0.156 + 0.987i)2-s + (1.57 − 0.717i)3-s + (−0.951 − 0.309i)4-s + (−0.919 − 2.03i)5-s + (0.462 + 1.66i)6-s + (2.93 + 2.93i)7-s + (0.453 − 0.891i)8-s + (1.96 − 2.26i)9-s + (2.15 − 0.589i)10-s + (0.613 + 0.844i)11-s + (−1.72 + 0.195i)12-s + (0.542 − 0.0859i)13-s + (−3.35 + 2.44i)14-s + (−2.91 − 2.55i)15-s + (0.809 + 0.587i)16-s + (−4.43 − 2.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (0.910 − 0.414i)3-s + (−0.475 − 0.154i)4-s + (−0.411 − 0.911i)5-s + (0.188 + 0.681i)6-s + (1.10 + 1.10i)7-s + (0.160 − 0.315i)8-s + (0.656 − 0.754i)9-s + (0.682 − 0.186i)10-s + (0.185 + 0.254i)11-s + (−0.496 + 0.0564i)12-s + (0.150 − 0.0238i)13-s + (−0.897 + 0.652i)14-s + (−0.751 − 0.659i)15-s + (0.202 + 0.146i)16-s + (−1.07 − 0.548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31709 + 0.199593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31709 + 0.199593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 3 | \( 1 + (-1.57 + 0.717i)T \) |
| 5 | \( 1 + (0.919 + 2.03i)T \) |
| good | 7 | \( 1 + (-2.93 - 2.93i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.613 - 0.844i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.542 + 0.0859i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (4.43 + 2.25i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (5.82 - 1.89i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.754 - 0.119i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.30 - 7.09i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.786 + 2.41i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.45 + 9.21i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (5.89 - 8.10i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.405 + 0.405i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.320 - 0.628i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.82 + 3.47i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.444 - 0.322i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.77 + 4.19i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.27 + 8.39i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-4.73 - 1.53i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.24 - 14.1i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-9.92 - 3.22i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.84 - 3.62i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (7.67 - 5.57i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.77 - 0.904i)T + (57.0 - 78.4i)T^{2} \) |
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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
−13.05280754207173958218270666308, −12.38762088101373048232154078999, −11.21858259801705519390119541092, −9.372976562803261473177880206746, −8.641945382937604271719660822658, −8.130844227857297302936408412682, −6.87167327224955846385873916492, −5.32961102583612422986850920169, −4.17794710949153050471871344835, −1.94297063113244880867497328944,
2.15198478215675269474542189812, 3.75487390949845220362358390764, 4.50571457295550849999170726758, 6.85676338293317586847793152818, 7.995325765551584832659774021247, 8.764295318252366471949492233491, 10.30591121520968285793344861149, 10.75212732972431204857273946522, 11.61315154908039878633903561184, 13.27574929980261141639381392392
