L(s) = 1 | + (0.226 − 0.392i)3-s − 2.54·7-s + (1.39 + 2.42i)9-s + 2.22·11-s + (−3.51 − 6.08i)13-s + (−1.27 + 2.21i)17-s + (−2.70 + 3.41i)19-s + (−0.575 + 0.997i)21-s + (4.01 + 6.95i)23-s + 2.62·27-s + (0.941 + 1.63i)29-s + 5.98·31-s + (0.503 − 0.872i)33-s + 2.86·37-s − 3.17·39-s + ⋯ |
L(s) = 1 | + (0.130 − 0.226i)3-s − 0.961·7-s + (0.465 + 0.806i)9-s + 0.670·11-s + (−0.973 − 1.68i)13-s + (−0.310 + 0.537i)17-s + (−0.620 + 0.784i)19-s + (−0.125 + 0.217i)21-s + (0.837 + 1.44i)23-s + 0.505·27-s + (0.174 + 0.302i)29-s + 1.07·31-s + (0.0876 − 0.151i)33-s + 0.470·37-s − 0.509·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.341452329\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341452329\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.70 - 3.41i)T \) |
good | 3 | \( 1 + (-0.226 + 0.392i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + (3.51 + 6.08i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.27 - 2.21i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.01 - 6.95i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.941 - 1.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 + (3.67 - 6.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.84 + 3.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.36 - 4.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.14 - 8.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.17 - 7.23i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.17 + 10.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.13 - 7.16i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.32 - 10.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.13 + 3.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + (-7.19 - 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.91 - 5.04i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.465524792590529586910965368209, −8.469907310309028585893291296110, −7.74458137180699814969003744564, −7.08729570098740716550863125815, −6.19095046644701767555144991619, −5.40234644690001867760861247455, −4.42379557412853289139977334673, −3.36005873496883264741079025758, −2.54581326813441706751142848398, −1.20509539312166740212599407145,
0.52500707971126980228598242998, 2.18033555829990323147410864727, 3.13059789724252463109510388978, 4.30782678980436673383415817842, 4.62035439326148836819363415498, 6.17343900733171035116394961141, 6.87781991340906955801142121921, 7.02811436034026860882736485723, 8.727824773009197420030170938640, 9.024215901478451985386075191109