| L(s) = 1 | + (0.135 − 1.40i)2-s + (2.54 + 1.84i)3-s + (−1.96 − 0.382i)4-s + (1.08 − 1.95i)5-s + (2.94 − 3.32i)6-s − 1.17i·7-s + (−0.804 + 2.71i)8-s + (2.12 + 6.54i)9-s + (−2.60 − 1.79i)10-s + (−3.77 − 1.22i)11-s + (−4.28 − 4.59i)12-s + (0.280 + 0.864i)13-s + (−1.65 − 0.159i)14-s + (6.37 − 2.95i)15-s + (3.70 + 1.50i)16-s + (3.27 + 4.50i)17-s + ⋯ |
| L(s) = 1 | + (0.0959 − 0.995i)2-s + (1.46 + 1.06i)3-s + (−0.981 − 0.191i)4-s + (0.486 − 0.873i)5-s + (1.20 − 1.35i)6-s − 0.444i·7-s + (−0.284 + 0.958i)8-s + (0.708 + 2.18i)9-s + (−0.823 − 0.567i)10-s + (−1.13 − 0.370i)11-s + (−1.23 − 1.32i)12-s + (0.0779 + 0.239i)13-s + (−0.442 − 0.0426i)14-s + (1.64 − 0.764i)15-s + (0.926 + 0.375i)16-s + (0.794 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64351 - 0.636201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64351 - 0.636201i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.135 + 1.40i)T \) |
| 5 | \( 1 + (-1.08 + 1.95i)T \) |
| good | 3 | \( 1 + (-2.54 - 1.84i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 1.17iT - 7T^{2} \) |
| 11 | \( 1 + (3.77 + 1.22i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.280 - 0.864i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.27 - 4.50i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.64 + 3.64i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.07 + 0.674i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.65 - 5.03i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.16 - 0.848i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.16 + 6.65i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 6.06i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + (3.12 - 4.30i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (10.5 + 7.68i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.397i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-14.7 - 4.78i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.45 + 1.78i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (1.58 + 1.14i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.01 - 1.30i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.23 - 1.62i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.65 + 5.56i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.806 + 2.48i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.171 - 0.235i)T + (-29.9 - 92.2i)T^{2} \) |
| 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
−12.76954838943886037323557259731, −10.96839057911845521608588145463, −10.26653019053691522472131888761, −9.498421377962725392643642826816, −8.621068667596093511742267042712, −7.995723371691472422492327577927, −5.40398294236525545345745233532, −4.41428666761591056872668850289, −3.38500521962143272282201301100, −2.04675084545750234017604318706,
2.31907847230298014550618504182, 3.48515769552994346845932863965, 5.54343037113265319433374355422, 6.67367018939605060566891899892, 7.62173038024048982832513652490, 8.128773915535984508138863405942, 9.357715471277908323720315761833, 10.12363037881869658668541934210, 12.10530335829015398417029349586, 13.04266906776694996046353783941
