L(s) = 1 | + (0.392 + 0.226i)3-s + (0.207 − 2.22i)5-s + 2.54i·7-s + (−1.39 − 2.42i)9-s + 2.22·11-s + (6.08 − 3.51i)13-s + (0.585 − 0.826i)15-s + (2.21 + 1.27i)17-s + (2.70 − 3.41i)19-s + (−0.575 + 0.997i)21-s + (−6.95 + 4.01i)23-s + (−4.91 − 0.921i)25-s − 2.62i·27-s + (−0.941 − 1.63i)29-s + 5.98·31-s + ⋯ |
L(s) = 1 | + (0.226 + 0.130i)3-s + (0.0925 − 0.995i)5-s + 0.961i·7-s + (−0.465 − 0.806i)9-s + 0.670·11-s + (1.68 − 0.973i)13-s + (0.151 − 0.213i)15-s + (0.537 + 0.310i)17-s + (0.620 − 0.784i)19-s + (−0.125 + 0.217i)21-s + (−1.44 + 0.837i)23-s + (−0.982 − 0.184i)25-s − 0.505i·27-s + (−0.174 − 0.302i)29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48114 - 0.417357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48114 - 0.417357i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.207 + 2.22i)T \) |
| 19 | \( 1 + (-2.70 + 3.41i)T \) |
good | 3 | \( 1 + (-0.392 - 0.226i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 2.54iT - 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + (-6.08 + 3.51i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 1.27i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (6.95 - 4.01i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.941 + 1.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + 2.86iT - 37T^{2} \) |
| 41 | \( 1 + (3.67 - 6.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.19 - 1.84i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.09 + 2.36i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.91 - 5.14i)T + (26.5 - 45.8i)T^{2} \) |
| 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 | \( 1 + (10.7 - 6.17i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.13 - 7.16i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (10.9 + 6.32i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.13 - 3.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 + (7.19 + 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.04 - 2.91i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.69785558784871154615884544303, −10.18504681513553471326806106493, −9.175318909424537444525625727733, −8.705036020016407088688794474243, −7.85959848165682692828819276060, −5.99809759686784438691843688981, −5.78543891709941455327520396731, −4.18805977591619457086579005892, −3.08897687340448786683246027040, −1.22107114699871479832816820134,
1.71052135781746079303649807617, 3.31762071764289047492525234846, 4.20452901050629795831178865826, 5.90712179409563849903672352775, 6.68838326055390880846131436031, 7.68896111427885849322569475900, 8.519232531342852732107405948366, 9.782734166253315140045971972882, 10.60625764267601167481305562298, 11.27295541511751541570467883995