| L(s) = 1 | + (−1.57 + 0.277i)2-s + (1.72 − 0.149i)3-s + (0.513 − 0.186i)4-s + (1.78 − 0.649i)5-s + (−2.67 + 0.713i)6-s + (−1.28 + 2.31i)7-s + (2.00 − 1.15i)8-s + (2.95 − 0.517i)9-s + (−2.62 + 1.51i)10-s + (0.432 − 1.18i)11-s + (0.858 − 0.399i)12-s + (0.326 + 0.898i)13-s + (1.37 − 3.99i)14-s + (2.97 − 1.38i)15-s + (−3.67 + 3.08i)16-s + (−2.00 − 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.195i)2-s + (0.996 − 0.0865i)3-s + (0.256 − 0.0934i)4-s + (0.797 − 0.290i)5-s + (−1.09 + 0.291i)6-s + (−0.485 + 0.874i)7-s + (0.710 − 0.410i)8-s + (0.985 − 0.172i)9-s + (−0.829 + 0.478i)10-s + (0.130 − 0.357i)11-s + (0.247 − 0.115i)12-s + (0.0906 + 0.249i)13-s + (0.368 − 1.06i)14-s + (0.769 − 0.358i)15-s + (−0.918 + 0.770i)16-s + (−0.486 − 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.980509 + 0.141550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.980509 + 0.141550i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 + 0.149i)T \) |
| 7 | \( 1 + (1.28 - 2.31i)T \) |
| good | 2 | \( 1 + (1.57 - 0.277i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.78 + 0.649i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.432 + 1.18i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 0.898i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.00 + 3.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.64 - 3.83i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.63 - 0.994i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.22 - 3.37i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.04 + 8.36i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + 7.99T + 37T^{2} \) |
| 41 | \( 1 + (3.01 - 1.09i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.111 - 0.632i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.03 + 2.92i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (11.3 + 6.54i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.65 + 1.38i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.05 - 2.89i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.371 + 2.10i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.154 + 0.0893i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 + (0.211 + 1.20i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (5.11 + 1.86i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.81 + 4.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.97 + 1.05i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.92031250299452353425852944865, −11.51561549252571809605344633814, −9.971582332217356166953723604539, −9.362918166591038970351882494856, −8.910073130463459635148036480140, −7.80462804819000670417700930140, −6.74721163463970484299846847729, −5.23337691865297504419392535730, −3.34526924068853770650333914024, −1.69964367941938754239151909366,
1.57320205016634672662086201686, 3.18803162628688912520696448542, 4.81143637002280306610859372378, 6.77405727930851215072026394776, 7.59588935621232223253780657991, 8.795869629099605330950809634682, 9.530341472112867282980540195091, 10.21868958975107971411504796445, 10.96195960375441655722200310281, 12.77194675912176138992913430151
