| L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (4.68 − 2.08i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (0.103 − 0.988i)11-s + (0.669 + 0.743i)12-s + (0.563 − 0.625i)13-s + (−0.535 − 5.09i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.0437 + 0.416i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.564 − 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (1.76 − 0.787i)7-s + (−0.286 + 0.207i)8-s + (0.304 + 0.135i)9-s + (−0.309 + 0.0657i)10-s + (0.0313 − 0.297i)11-s + (0.193 + 0.214i)12-s + (0.156 − 0.173i)13-s + (−0.143 − 1.36i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.0106 + 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.790349 - 1.39999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790349 - 1.39999i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (2.86 - 4.77i)T \) |
| good | 7 | \( 1 + (-4.68 + 2.08i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.103 + 0.988i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.563 + 0.625i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0437 - 0.416i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.95 - 3.28i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.99 + 2.90i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.87 + 8.84i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.56 - 1.82i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.72 + 1.91i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.166 - 0.513i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.24 + 2.33i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (10.1 + 2.14i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + (2.28 + 3.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.50 + 4.23i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (1.18 - 11.3i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (1.14 + 10.9i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 0.332i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-7.18 - 5.21i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 7.29i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−10.19814175171951747163841814071, −8.915397912287132569290863818679, −8.090432234713565216757416422431, −7.44622474149654388911420985685, −6.13456953446804139400428500582, −5.04382981998479345112385064107, −4.59690641280980269077268305077, −3.51392118246282672775604841693, −1.81416447456031528030123142554, −0.868759754212937766771860176755,
1.55588969238159545285261200564, 3.12316872179163639231221821570, 4.61080618051926127631247984066, 5.03900230634226325648496856427, 5.91196500776438328778774315528, 7.04922020426571782483867899512, 7.63422969160216835141811657047, 8.614913194198349642726469520430, 9.268313254583683965774541146668, 10.52004689451933389916960206473
