L(s) = 1 | + (1.58 + 0.917i)2-s + (0.682 + 1.18i)4-s + (−1.90 + 3.29i)5-s + (2.11 + 1.58i)7-s − 1.16i·8-s + (−6.04 + 3.49i)10-s + (1.00 − 0.582i)11-s − 5.54i·13-s + (1.90 + 4.46i)14-s + (2.43 − 4.21i)16-s + (1.58 + 2.75i)17-s + (−0.546 − 0.315i)19-s − 5.19·20-s + 2.13·22-s + (−1.52 − 0.880i)23-s + ⋯ |
L(s) = 1 | + (1.12 + 0.648i)2-s + (0.341 + 0.590i)4-s + (−0.851 + 1.47i)5-s + (0.799 + 0.600i)7-s − 0.412i·8-s + (−1.91 + 1.10i)10-s + (0.304 − 0.175i)11-s − 1.53i·13-s + (0.508 + 1.19i)14-s + (0.608 − 1.05i)16-s + (0.385 + 0.667i)17-s + (−0.125 − 0.0724i)19-s − 1.16·20-s + 0.455·22-s + (−0.318 − 0.183i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47613 + 1.14995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47613 + 1.14995i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.11 - 1.58i)T \) |
good | 2 | \( 1 + (-1.58 - 0.917i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.90 - 3.29i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 0.582i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.54iT - 13T^{2} \) |
| 17 | \( 1 + (-1.58 - 2.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.546 + 0.315i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.52 + 0.880i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.83iT - 29T^{2} \) |
| 31 | \( 1 + (3.70 - 2.13i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + (1.27 - 2.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0627 + 0.0362i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.49 + 6.04i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.00 - 4.04i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (-4.84 + 2.79i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.54 - 14.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + (-5.77 + 10.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.688iT - 97T^{2} \) |
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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
−12.83776050303653185253128180420, −11.95732362607597373393980263911, −10.99881195876584158481961985532, −10.10563375861444202699405975483, −8.245634178668131832678468101777, −7.47628633510653001601674173263, −6.34700139638542543082677709435, −5.43096866486366871921976203552, −4.01017293493023535629077665395, −2.96794080793941839423489212343,
1.63649846551935434181554606907, 3.86599776265850199085720843752, 4.48149939732343399772445372764, 5.34873161510686683829534024466, 7.25863314144567465941719138208, 8.371310953175363849939441258240, 9.272786167368186191706565457624, 10.94539665148817168489856545570, 11.79303635901013486029361482978, 12.19275218204896694933437796099