L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.664 + 1.15i)3-s + (−0.499 − 0.866i)4-s + (−0.664 − 1.15i)6-s − 2.32·7-s + 0.999·8-s + (0.616 + 1.06i)9-s + 6.39·11-s + 1.32·12-s + (−0.429 − 0.743i)13-s + (1.16 − 2.01i)14-s + (−0.5 + 0.866i)16-s + (2.34 − 4.06i)17-s − 1.23·18-s + (3.75 − 2.21i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.383 + 0.664i)3-s + (−0.249 − 0.433i)4-s + (−0.271 − 0.469i)6-s − 0.880·7-s + 0.353·8-s + (0.205 + 0.355i)9-s + 1.92·11-s + 0.383·12-s + (−0.119 − 0.206i)13-s + (0.311 − 0.539i)14-s + (−0.125 + 0.216i)16-s + (0.568 − 0.985i)17-s − 0.290·18-s + (0.860 − 0.509i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.707596 + 0.886363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.707596 + 0.886363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.75 + 2.21i)T \) |
good | 3 | \( 1 + (0.664 - 1.15i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 13 | \( 1 + (0.429 + 0.743i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.34 + 4.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 3.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.21 - 3.82i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + (1.84 - 3.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.56 - 6.17i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.59 - 6.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.10 - 5.37i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.09 + 5.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.01 - 6.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.45 - 4.24i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.10 - 1.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.32 + 4.03i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.79 - 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 + (5.64 + 9.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.67 + 9.82i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.710125076867203109153913987615, −9.701175661941259452051508838605, −8.880329967427364126458837084941, −7.53362540538182589174229862626, −6.94692540853130454079853094040, −6.00862640297876360983006241679, −5.15305480631211983284809331285, −4.18823243861168502946957270320, −3.14595106809853061389356045402, −1.16438413315450881630524331328,
0.803628700261515346244694529027, 1.85860802354086149075259747756, 3.50379208678512685292058752622, 4.01326985980045368255700045302, 5.69981293151174283884697345229, 6.54901645766415467793415252167, 7.08730719355722403980478700752, 8.224493629783686758745004218951, 9.289918592516277336506470607371, 9.631519146175217847797250875454