L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.34 − 1.22i)7-s − 0.999i·8-s + (−2.03 − 1.17i)11-s + 4.64i·13-s + (−2.64 + 0.107i)14-s + (−0.5 − 0.866i)16-s + (−2.28 + 3.95i)17-s + (0.491 − 0.283i)19-s − 2.35·22-s + (5.04 − 2.91i)23-s + (2.32 + 4.02i)26-s + (−2.23 + 1.41i)28-s + 2.55i·29-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.885 − 0.464i)7-s − 0.353i·8-s + (−0.615 − 0.355i)11-s + 1.28i·13-s + (−0.706 + 0.0287i)14-s + (−0.125 − 0.216i)16-s + (−0.553 + 0.958i)17-s + (0.112 − 0.0650i)19-s − 0.502·22-s + (1.05 − 0.607i)23-s + (0.455 + 0.789i)26-s + (−0.422 + 0.267i)28-s + 0.474i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.385209669\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385209669\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.34 + 1.22i)T \) |
good | 11 | \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.64iT - 13T^{2} \) |
| 17 | \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.491 + 0.283i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.04 + 2.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.55iT - 29T^{2} \) |
| 31 | \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.63 - 8.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + (-3.15 - 5.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 6.07i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.67 - 2.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.02iT - 71T^{2} \) |
| 73 | \( 1 + (-7.11 - 4.10i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.13 - 7.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.171T + 83T^{2} \) |
| 89 | \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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
−8.936921760481493462039809644610, −8.103679575565471546696899391164, −6.93595676392368715623652253596, −6.64427921784762187786783567131, −5.78336776851344744475408557857, −4.80337589758956622749306225994, −4.11088417834676815495736752904, −3.27383281292110736983139418071, −2.43858155800744158202956994845, −1.23073506466915327065917580952,
0.34824325723573838997817622041, 2.22815964022295487314061009895, 3.02700440910445841536465449834, 3.69624027699207786119685211172, 5.01327233025602043736626325708, 5.34051815024111638002127753820, 6.22370094108809353614437665548, 7.05065493395910546178552190735, 7.58312466329767653537291953435, 8.494426157767703898564192606065