L(s) = 1 | + (1.69 + 0.346i)3-s + (1.79 − 3.10i)5-s + (−2.56 + 0.634i)7-s + (2.75 + 1.17i)9-s + (−0.200 + 0.115i)11-s + (1.16 + 0.673i)13-s + (4.11 − 4.64i)15-s + 7.94·17-s − 3.06i·19-s + (−4.57 + 0.185i)21-s + (−4.87 − 2.81i)23-s + (−3.91 − 6.78i)25-s + (4.27 + 2.95i)27-s + (−2.33 + 1.34i)29-s + (1.85 + 1.07i)31-s + ⋯ |
L(s) = 1 | + (0.979 + 0.200i)3-s + (0.801 − 1.38i)5-s + (−0.970 + 0.239i)7-s + (0.919 + 0.392i)9-s + (−0.0604 + 0.0348i)11-s + (0.323 + 0.186i)13-s + (1.06 − 1.19i)15-s + 1.92·17-s − 0.702i·19-s + (−0.999 + 0.0403i)21-s + (−1.01 − 0.586i)23-s + (−0.783 − 1.35i)25-s + (0.822 + 0.568i)27-s + (−0.433 + 0.250i)29-s + (0.332 + 0.192i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04487 - 0.533407i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04487 - 0.533407i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.346i)T \) |
| 7 | \( 1 + (2.56 - 0.634i)T \) |
good | 5 | \( 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.200 - 0.115i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 0.673i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7.94T + 17T^{2} \) |
| 19 | \( 1 + 3.06iT - 19T^{2} \) |
| 23 | \( 1 + (4.87 + 2.81i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.33 - 1.34i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.85 - 1.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + (0.813 - 1.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.927 + 1.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0396 + 0.0686i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 + (6.48 - 11.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.729 + 0.420i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.05 - 8.75i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.47iT - 71T^{2} \) |
| 73 | \( 1 + 7.97iT - 73T^{2} \) |
| 79 | \( 1 + (3.30 + 5.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.41 - 11.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + (13.1 - 7.58i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.36761351810986428853908051081, −9.797641232303216969301497560168, −9.055882936174746190416819722812, −8.478607282540438618455005025964, −7.38940561860672568288025127758, −6.06645309514888396167124364819, −5.17364809334650547575232398142, −3.99282004226477122740161483082, −2.80109073370153259809820439919, −1.38301720439243071305615472362,
1.85718735129666186894263518830, 3.19560636534850572322270019872, 3.58594895883412851137395692877, 5.70140920607845125315159442241, 6.47766651394079791138298026822, 7.37505996133296488817948256830, 8.145945328213616117527967819281, 9.598043691744792247630780953932, 9.926490878861464010851191977618, 10.60048739484787449167520964380