L(s) = 1 | + (1.34 − 1.60i)2-s + (0.0648 − 1.73i)3-s + (−0.412 − 2.34i)4-s + (0.275 − 0.231i)5-s + (−2.68 − 2.43i)6-s + (2.18 + 1.48i)7-s + (−0.682 − 0.393i)8-s + (−2.99 − 0.224i)9-s − 0.753i·10-s + (−4.02 + 4.79i)11-s + (−4.07 + 0.562i)12-s + (−0.429 + 1.18i)13-s + (5.32 − 1.50i)14-s + (−0.382 − 0.492i)15-s + (2.91 − 1.06i)16-s − 0.937·17-s + ⋯ |
L(s) = 1 | + (0.950 − 1.13i)2-s + (0.0374 − 0.999i)3-s + (−0.206 − 1.17i)4-s + (0.123 − 0.103i)5-s + (−1.09 − 0.992i)6-s + (0.826 + 0.562i)7-s + (−0.241 − 0.139i)8-s + (−0.997 − 0.0748i)9-s − 0.238i·10-s + (−1.21 + 1.44i)11-s + (−1.17 + 0.162i)12-s + (−0.119 + 0.327i)13-s + (1.42 − 0.402i)14-s + (−0.0988 − 0.127i)15-s + (0.729 − 0.265i)16-s − 0.227·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.999894 - 1.62285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999894 - 1.62285i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0648 + 1.73i)T \) |
| 7 | \( 1 + (-2.18 - 1.48i)T \) |
good | 2 | \( 1 + (-1.34 + 1.60i)T + (-0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.275 + 0.231i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (4.02 - 4.79i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.429 - 1.18i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + 0.937T + 17T^{2} \) |
| 19 | \( 1 + 7.64iT - 19T^{2} \) |
| 23 | \( 1 + (0.584 - 1.60i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.731 + 2.00i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.71 + 1.00i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.66 + 2.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 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 | \( 1 + (-0.876 + 4.96i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.01 + 1.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.00 - 2.18i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (10.3 + 1.81i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.222 - 0.186i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.18 - 2.41i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.19 - 1.84i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.90 - 6.63i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 2.20i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + (7.91 + 1.39i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.24150341069359170268976942391, −11.62107751493246212924509815194, −10.77427388189753592642336534957, −9.440945731418004088750740145775, −8.067818510300314053208373596590, −7.08324249926027893509945272360, −5.40238711042154922675220161073, −4.66995675603871063238523977797, −2.68069300887197569354552504154, −1.87886077041601464381543998017,
3.28032405396210625880611920173, 4.48166504837461339191298881305, 5.40536287791748421031166734680, 6.23468009336392138126580024874, 7.957146340466866956713972284117, 8.308032526988853205876890354183, 10.24025887286840814169016381775, 10.68672317346131645756414247450, 12.02191921884807073847841468139, 13.45560645780329636845273176711