L(s) = 1 | + (1.97 + 1.13i)2-s + (1.59 + 2.75i)4-s + (−0.717 − 1.24i)5-s + (2.16 + 1.52i)7-s + 2.69i·8-s − 3.26i·10-s + (−2.80 − 1.61i)11-s + (−4.43 + 2.55i)13-s + (2.52 + 5.46i)14-s + (0.119 − 0.207i)16-s + 1.09·17-s − 4.48i·19-s + (2.28 − 3.95i)20-s + (−3.68 − 6.37i)22-s + (−3.47 + 2.00i)23-s + ⋯ |
L(s) = 1 | + (1.39 + 0.804i)2-s + (0.795 + 1.37i)4-s + (−0.320 − 0.555i)5-s + (0.817 + 0.576i)7-s + 0.951i·8-s − 1.03i·10-s + (−0.844 − 0.487i)11-s + (−1.22 + 0.709i)13-s + (0.675 + 1.46i)14-s + (0.0298 − 0.0517i)16-s + 0.264·17-s − 1.02i·19-s + (0.510 − 0.883i)20-s + (−0.784 − 1.35i)22-s + (−0.723 + 0.417i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97704 + 1.03564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97704 + 1.03564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.16 - 1.52i)T \) |
good | 2 | \( 1 + (-1.97 - 1.13i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.717 + 1.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.80 + 1.61i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.43 - 2.55i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.09T + 17T^{2} \) |
| 19 | \( 1 + 4.48iT - 19T^{2} \) |
| 23 | \( 1 + (3.47 - 2.00i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.02 + 0.593i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 1.87i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.239T + 37T^{2} \) |
| 41 | \( 1 + (-3.71 - 6.43i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.11 - 3.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.01iT - 53T^{2} \) |
| 59 | \( 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.82 - 1.63i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.82iT - 71T^{2} \) |
| 73 | \( 1 + 7.31iT - 73T^{2} \) |
| 79 | \( 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.45 - 9.44i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + (2.69 + 1.55i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.82044507812302554633952976846, −12.05929674419499887300246292951, −11.28198629767966383160236363855, −9.618132102576302245054999150946, −8.250795887119348265482098227398, −7.49270320227827133549069104588, −6.16096397987087651524095263886, −5.02641159710826919225303779519, −4.49587522070343263889297810480, −2.70897374221943998806039585883,
2.17910684348331092153140832136, 3.48679074714212651949736984927, 4.69099158626331895663462462267, 5.54867858174274211105772475266, 7.18774944894088821947460012875, 8.083451446390346511575896714812, 10.22780628863034258053843949314, 10.52223317160537809516397193544, 11.71949585910116902473558136567, 12.37020126852564543296614955014