L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s − 2·7-s + 0.999·8-s + (1 + 1.73i)9-s − 3·11-s + 0.999·12-s + (3 + 5.19i)13-s + (1 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s − 2·18-s + (3.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s − 0.755·7-s + 0.353·8-s + (0.333 + 0.577i)9-s − 0.904·11-s + 0.288·12-s + (0.832 + 1.44i)13-s + (0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s − 0.471·18-s + (0.802 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.122636 - 0.381867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122636 - 0.381867i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.5 - 7.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.5 - 12.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.36859661640223438742265491932, −9.723062748053496413028711491741, −8.925555673642421527702407336663, −8.016702852100782720064155637798, −7.12708564082600825366052882373, −6.31965801127772253482119207790, −5.39971952835208150949969959237, −4.54638968321674808446313788390, −3.47455028564427821478408444633, −1.86194946388814138926680489730,
0.21899355817458887370701028620, 1.55806517875272411449924869470, 3.11755097200733550119288067858, 3.68746108087346518524154159102, 5.35379623129920964278335808955, 6.00537553320877504752555515351, 7.20960670757774901493112066904, 7.80166390321345304139152639750, 8.809603457363235577501433377780, 9.733230303290900359360727722307