| L(s) = 1 | + (0.484 + 1.32i)2-s + (1.73 + 0.0238i)3-s + (−1.52 + 1.28i)4-s + (2.22 − 3.33i)5-s + (0.808 + 2.31i)6-s + (0.140 + 0.338i)7-s + (−2.45 − 1.40i)8-s + (2.99 + 0.0824i)9-s + (5.51 + 1.34i)10-s + (−4.16 + 0.827i)11-s + (−2.67 + 2.19i)12-s + (−1.81 + 1.21i)13-s + (−0.381 + 0.350i)14-s + (3.93 − 5.72i)15-s + (0.679 − 3.94i)16-s + (−2.34 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.342 + 0.939i)2-s + (0.999 + 0.0137i)3-s + (−0.764 + 0.644i)4-s + (0.996 − 1.49i)5-s + (0.329 + 0.943i)6-s + (0.0530 + 0.127i)7-s + (−0.867 − 0.497i)8-s + (0.999 + 0.0274i)9-s + (1.74 + 0.424i)10-s + (−1.25 + 0.249i)11-s + (−0.773 + 0.633i)12-s + (−0.502 + 0.335i)13-s + (−0.102 + 0.0936i)14-s + (1.01 − 1.47i)15-s + (0.169 − 0.985i)16-s + (−0.568 + 0.568i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67980 + 0.720315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67980 + 0.720315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.484 - 1.32i)T \) |
| 3 | \( 1 + (-1.73 - 0.0238i)T \) |
| good | 5 | \( 1 + (-2.22 + 3.33i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.140 - 0.338i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.16 - 0.827i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.81 - 1.21i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (2.34 - 2.34i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.58 - 3.73i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (0.561 - 1.35i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (0.511 - 2.57i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 + (-2.65 + 3.96i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-5.19 - 2.15i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.439 - 0.0874i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-2.23 - 2.23i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.03 + 5.20i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (6.44 + 4.30i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.26 + 11.3i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-3.80 - 0.757i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-3.54 + 1.46i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (8.70 + 3.60i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.50 - 2.50i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.65 + 2.47i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (11.1 - 4.63i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 10.0iT - 97T^{2} \) |
| 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
−12.83063867094100162113746026184, −12.51869327159977588739699310665, −10.19887757661727426869586901391, −9.345796175626186133444176418786, −8.498005998433998872069486716695, −7.86153785198855842254889938196, −6.32480885820092394930267005383, −5.12313065274748319707376705801, −4.25635429342559960006271556901, −2.21466933438216299270468024900,
2.49877522311618626904913577777, 2.73943228914962099846244533380, 4.50357328707296818519171700658, 6.04642154430400808162827726089, 7.28255463612415746927107528277, 8.647483960289427484667246262036, 9.826631502382829440501215344120, 10.39235434549231049516294157434, 11.12807248059424609925385572842, 12.70549717322733672457793871584
