| L(s) = 1 | + (1.40 − 0.124i)2-s + (−0.392 + 1.68i)3-s + (1.96 − 0.350i)4-s + (2.20 + 0.380i)5-s + (−0.343 + 2.42i)6-s − 0.0822i·7-s + (2.73 − 0.738i)8-s + (−2.69 − 1.32i)9-s + (3.15 + 0.262i)10-s + (−2.26 + 1.64i)11-s + (−0.182 + 3.45i)12-s + (−3.85 − 2.80i)13-s + (−0.0102 − 0.115i)14-s + (−1.50 + 3.56i)15-s + (3.75 − 1.38i)16-s + (3.83 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (0.996 − 0.0879i)2-s + (−0.226 + 0.973i)3-s + (0.984 − 0.175i)4-s + (0.985 + 0.170i)5-s + (−0.140 + 0.990i)6-s − 0.0310i·7-s + (0.965 − 0.261i)8-s + (−0.897 − 0.441i)9-s + (0.996 + 0.0829i)10-s + (−0.681 + 0.495i)11-s + (−0.0525 + 0.998i)12-s + (−1.06 − 0.777i)13-s + (−0.00273 − 0.0309i)14-s + (−0.389 + 0.921i)15-s + (0.938 − 0.345i)16-s + (0.929 − 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22660 + 0.779372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22660 + 0.779372i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.124i)T \) |
| 3 | \( 1 + (0.392 - 1.68i)T \) |
| 5 | \( 1 + (-2.20 - 0.380i)T \) |
| good | 7 | \( 1 + 0.0822iT - 7T^{2} \) |
| 11 | \( 1 + (2.26 - 1.64i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.85 + 2.80i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.83 + 1.24i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.03 - 0.661i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.93 - 1.40i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.40 - 1.10i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.91 - 0.947i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.33 + 5.32i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.77 - 6.57i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.76iT - 43T^{2} \) |
| 47 | \( 1 + (-0.650 + 2.00i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.15 + 2.32i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.56 + 5.49i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.43 - 1.77i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-12.2 + 3.97i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 3.42i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.23 - 0.898i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.51 - 1.46i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.96 - 9.13i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.21 - 8.55i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.56 - 7.89i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.04750346371552419581410044981, −10.63814710109599438447143998099, −10.30576245653894662747194283921, −9.445823687396431197744009294467, −7.83918352123552403230320159234, −6.61064085099658751829614817081, −5.36553039123121320785378466381, −5.07282128368124085829338520947, −3.49928529351613145809773175592, −2.35924924566440180895003520880,
1.81222085570209628760601115503, 2.88637888973724154554869550019, 4.81156139854972753445760185836, 5.70254387783375896228815535367, 6.51069249316697588841156488030, 7.47413027093643409315956909289, 8.536919167239741503307818250555, 10.02513478977972741103692968556, 10.92456895241619618100819327764, 12.09202060194362611932362727501
