| L(s) = 1 | + (−0.453 + 0.891i)2-s + (−0.194 − 1.23i)3-s + (−0.587 − 0.809i)4-s + (0.793 + 2.09i)5-s + (1.18 + 0.384i)6-s + (2.94 + 0.466i)7-s + (0.987 − 0.156i)8-s + (1.37 − 0.447i)9-s + (−2.22 − 0.242i)10-s + (3.28 + 0.461i)11-s + (−0.880 + 0.880i)12-s + (−5.36 − 2.73i)13-s + (−1.75 + 2.41i)14-s + (2.41 − 1.38i)15-s + (−0.309 + 0.951i)16-s + (−5.89 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (−0.112 − 0.710i)3-s + (−0.293 − 0.404i)4-s + (0.354 + 0.934i)5-s + (0.483 + 0.157i)6-s + (1.11 + 0.176i)7-s + (0.349 − 0.0553i)8-s + (0.459 − 0.149i)9-s + (−0.702 − 0.0765i)10-s + (0.990 + 0.139i)11-s + (−0.254 + 0.254i)12-s + (−1.48 − 0.757i)13-s + (−0.468 + 0.645i)14-s + (0.624 − 0.357i)15-s + (−0.0772 + 0.237i)16-s + (−1.42 + 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.934862 + 0.227300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.934862 + 0.227300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 5 | \( 1 + (-0.793 - 2.09i)T \) |
| 11 | \( 1 + (-3.28 - 0.461i)T \) |
| good | 3 | \( 1 + (0.194 + 1.23i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-2.94 - 0.466i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (5.36 + 2.73i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (5.89 - 3.00i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.55 + 1.13i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.606 - 0.606i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.56 + 2.58i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.337 + 1.03i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.15 + 7.26i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (4.84 - 6.66i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (3.90 - 3.90i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.22 + 1.14i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (2.32 - 4.55i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (4.39 + 6.04i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.11 + 1.33i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.06 - 1.06i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.18 + 3.65i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.108 - 0.684i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.26 + 3.88i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.449 + 0.881i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 + (-4.45 - 2.26i)T + (57.0 + 78.4i)T^{2} \) |
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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
−13.96304753888820942908091128343, −12.81574393208170300456508129109, −11.63541161465905828694832127530, −10.54213473497506057944072618326, −9.396505584579768187874430544747, −7.993297231209710044529239501159, −7.07126917044040365878980628342, −6.20990722694194931643716862881, −4.57918463740995026496374415899, −2.02184318564929023246610069631,
1.83239860429422666909116167667, 4.39712545363086446918916119072, 4.85797988567735746212887149780, 7.06951785235274130986280694766, 8.615482270849420762598612727692, 9.365997104560289100506955734993, 10.35100391894433437127694341414, 11.51293791996693007940451233697, 12.25725738463827060549223272260, 13.56106617881896960541243805802
