L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.189 + 2.63i)7-s − 0.999i·8-s + (2.55 − 1.47i)11-s + 3.93i·13-s + (1.48 − 2.19i)14-s + (−0.5 + 0.866i)16-s + (0.199 + 0.346i)17-s + (0.0305 + 0.0176i)19-s − 2.94·22-s + (−3.23 − 1.86i)23-s + (1.96 − 3.40i)26-s + (−2.38 + 1.15i)28-s + 8.89i·29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0716 + 0.997i)7-s − 0.353i·8-s + (0.769 − 0.444i)11-s + 1.09i·13-s + (0.396 − 0.585i)14-s + (−0.125 + 0.216i)16-s + (0.0484 + 0.0839i)17-s + (0.00700 + 0.00404i)19-s − 0.628·22-s + (−0.673 − 0.389i)23-s + (0.385 − 0.667i)26-s + (−0.449 + 0.218i)28-s + 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9604714722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9604714722\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.189 - 2.63i)T \) |
good | 11 | \( 1 + (-2.55 + 1.47i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.93iT - 13T^{2} \) |
| 17 | \( 1 + (-0.199 - 0.346i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0305 - 0.0176i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.23 + 1.86i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-0.717 + 0.414i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.96 - 6.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 + (-2.90 + 5.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.72 + 2.14i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.78 - 4.82i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.97 + 5.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.25 + 10.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.93iT - 71T^{2} \) |
| 73 | \( 1 + (-0.297 + 0.171i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.15 - 7.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + (3.08 - 5.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.6iT - 97T^{2} \) |
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
−8.849785654079806902784204198575, −8.520431101661453294749734185961, −7.47265838682520022577210721126, −6.62916956839217891096604533340, −6.08118481757207300746466329137, −5.05638529689564962963254232305, −4.06810800868188482742715656300, −3.18715458234126220354981940192, −2.20894627774537117853757703480, −1.31659106807686206814217734119,
0.38830614694181141914515757695, 1.44616642469027448073438007462, 2.67824660660693284817720585860, 3.86118122374679568593480465307, 4.48885688740841652859696381316, 5.71069829243081729408260651299, 6.20240419590698298210812274915, 7.37300210489151553178339873056, 7.47094854580024458151779143394, 8.403096654444892770476015731747