L(s) = 1 | + (1.05 − 0.610i)2-s + (−1.97 + 1.14i)3-s + (−0.253 + 0.439i)4-s + (−1.39 + 2.41i)6-s − 1.28i·7-s + 3.06i·8-s + (1.11 − 1.92i)9-s + 0.285·11-s − 1.15i·12-s + (−4.33 − 2.5i)13-s + (−0.785 − 1.35i)14-s + (1.36 + 2.36i)16-s + (−5.40 + 3.11i)17-s − 2.71i·18-s + (−2.92 + 3.22i)19-s + ⋯ |
L(s) = 1 | + (0.748 − 0.431i)2-s + (−1.14 + 0.659i)3-s + (−0.126 + 0.219i)4-s + (−0.569 + 0.987i)6-s − 0.485i·7-s + 1.08i·8-s + (0.370 − 0.641i)9-s + 0.0859·11-s − 0.334i·12-s + (−1.20 − 0.693i)13-s + (−0.209 − 0.363i)14-s + (0.341 + 0.590i)16-s + (−1.30 + 0.756i)17-s − 0.639i·18-s + (−0.671 + 0.740i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0774772 + 0.418922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0774772 + 0.418922i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.92 - 3.22i)T \) |
good | 2 | \( 1 + (-1.05 + 0.610i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.97 - 1.14i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 1.28iT - 7T^{2} \) |
| 11 | \( 1 - 0.285T + 11T^{2} \) |
| 13 | \( 1 + (4.33 + 2.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.40 - 3.11i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.53 + 2.61i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.642 + 1.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (-0.420 - 0.728i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.28 + 2.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.96 + 2.86i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 6.18i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.86 - 4.96i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.22 - 3.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.853 - 0.492i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.46 - 2.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.661 + 0.382i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.72 + 13.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.66iT - 83T^{2} \) |
| 89 | \( 1 + (8.01 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 5.87i)T + (48.5 - 84.0i)T^{2} \) |
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
−11.56514996166504089215770034041, −10.46067128634188558673210746034, −10.25516430134944401276622321390, −8.729703642535555454192138260761, −7.77475961793857102717930458514, −6.42819116472011528814630282948, −5.49125149182208956423409948696, −4.53509842726230043866139650438, −3.99934317404993752019701760664, −2.41897150486658590951761879988,
0.22081658799893639279688565388, 2.19399073881162927565203937138, 4.21063509553018479124990555997, 5.08402879534860973084003739608, 5.85760016908693035345034342286, 6.75882242763270327330455662604, 7.24947221734901429978213369810, 8.941775750898559417401580050820, 9.696271247022186578551183547897, 10.93116931835989526776793032931