L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (−1.73 + 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.5 − 2.59i)11-s + (0.866 + 0.499i)12-s + 5i·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (2.5 − 4.33i)19-s + (−0.499 + 2.59i)21-s + 3i·22-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (−0.654 + 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.452 − 0.783i)11-s + (0.249 + 0.144i)12-s + 1.38i·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.204 + 0.117i)18-s + (0.573 − 0.993i)19-s + (−0.109 + 0.566i)21-s + 0.639i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.238619274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238619274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.79 - 4.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + (-2.59 - 1.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.677492838358494712612640222437, −8.998784540809379618849885936098, −8.718766016554221046082468019169, −7.46749410862873346855352302644, −6.83668335869530616314717622520, −5.86070959764679292981238071897, −4.62854885301692593513860835370, −3.18760925455184397667499783025, −2.67691697405132522064789992797, −1.22134477483076910399290262150,
0.74027399167871410108097694722, 2.47701985593217600438889990559, 3.48043691848987554180668374927, 4.65900193572421154527749653667, 5.66446051003091680172721572848, 6.71642180873984443793157184802, 7.60772875261231154516908676574, 8.037954885766469615236759417321, 9.141947678278473600374527735104, 9.889656149684426643687844228072