| L(s) = 1 | + (−2.06 − 0.363i)2-s + (−1.46 − 0.921i)3-s + (2.23 + 0.812i)4-s + (0.338 + 0.123i)5-s + (2.68 + 2.43i)6-s + (1.84 + 1.89i)7-s + (−0.682 − 0.393i)8-s + (1.30 + 2.70i)9-s + (−0.652 − 0.376i)10-s + (−2.14 − 5.88i)11-s + (−2.52 − 3.25i)12-s + (0.429 − 1.18i)13-s + (−3.11 − 4.57i)14-s + (−0.382 − 0.492i)15-s + (−2.37 − 1.99i)16-s + (−0.468 + 0.811i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 − 0.256i)2-s + (−0.846 − 0.532i)3-s + (1.11 + 0.406i)4-s + (0.151 + 0.0551i)5-s + (1.09 + 0.992i)6-s + (0.697 + 0.716i)7-s + (−0.241 − 0.139i)8-s + (0.433 + 0.901i)9-s + (−0.206 − 0.119i)10-s + (−0.645 − 1.77i)11-s + (−0.729 − 0.938i)12-s + (0.119 − 0.327i)13-s + (−0.831 − 1.22i)14-s + (−0.0988 − 0.127i)15-s + (−0.594 − 0.498i)16-s + (−0.113 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338147 - 0.298219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338147 - 0.298219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.46 + 0.921i)T \) |
| 7 | \( 1 + (-1.84 - 1.89i)T \) |
| good | 2 | \( 1 + (2.06 + 0.363i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.338 - 0.123i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (2.14 + 5.88i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.429 + 1.18i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.468 - 0.811i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.61 + 3.82i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.68 + 0.296i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.731 + 2.00i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.98 + 5.45i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 3.33T + 37T^{2} \) |
| 41 | \( 1 + (-8.66 - 3.15i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.688 - 3.90i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.74 + 1.72i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.01 + 1.73i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.89 + 4.10i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (3.57 + 9.82i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.0504 + 0.286i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.18 - 2.41i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 + (-1.79 + 10.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (6.06 - 2.20i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.56 - 4.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.91 - 1.39i)T + (91.1 + 33.1i)T^{2} \) |
| 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
−11.71569896175947255168623417351, −11.30339988238200012531178214175, −10.51128655625087553300088481874, −9.313471436327275985537347517461, −8.231711084185181120261147622424, −7.64603286670283368912890780118, −6.11716047759204283896164760536, −5.16905573536188679539422449342, −2.52157115201602697324636529991, −0.807510199935321077630258407787,
1.45571919458140115084979224696, 4.26342216572203285374558124711, 5.40619482184326321464523890817, 7.13021326749334228498722207111, 7.50723117812342329548970924196, 9.032234996229918349365396135194, 9.956283289785239996578616852285, 10.41748322170447305599018719607, 11.44367270444677736427516935029, 12.43343071670128546568538836962
