| L(s) = 1 | + (1.40 + 0.144i)2-s + (−0.926 + 1.46i)3-s + (1.95 + 0.407i)4-s + (−2.17 + 0.501i)5-s + (−1.51 + 1.92i)6-s + 2.67i·7-s + (2.69 + 0.855i)8-s + (−1.28 − 2.71i)9-s + (−3.13 + 0.390i)10-s + (1.85 + 1.34i)11-s + (−2.40 + 2.48i)12-s + (−2.84 + 2.06i)13-s + (−0.387 + 3.76i)14-s + (1.28 − 3.65i)15-s + (3.66 + 1.59i)16-s + (−1.64 − 0.533i)17-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.102i)2-s + (−0.534 + 0.845i)3-s + (0.979 + 0.203i)4-s + (−0.974 + 0.224i)5-s + (−0.618 + 0.785i)6-s + 1.01i·7-s + (0.953 + 0.302i)8-s + (−0.428 − 0.903i)9-s + (−0.992 + 0.123i)10-s + (0.558 + 0.405i)11-s + (−0.695 + 0.718i)12-s + (−0.789 + 0.573i)13-s + (−0.103 + 1.00i)14-s + (0.331 − 0.943i)15-s + (0.917 + 0.398i)16-s + (−0.398 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07829 + 1.30036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07829 + 1.30036i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.144i)T \) |
| 3 | \( 1 + (0.926 - 1.46i)T \) |
| 5 | \( 1 + (2.17 - 0.501i)T \) |
| good | 7 | \( 1 - 2.67iT - 7T^{2} \) |
| 11 | \( 1 + (-1.85 - 1.34i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.84 - 2.06i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.64 + 0.533i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.93 - 2.25i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (6.63 + 4.81i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.87 + 1.90i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.24 - 0.406i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.58 + 5.50i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.57 + 6.30i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.46iT - 43T^{2} \) |
| 47 | \( 1 + (-1.80 - 5.55i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.712 + 0.231i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.338 + 0.245i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.39 + 4.64i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.682 - 0.221i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.00682 - 0.0210i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 3.39i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (9.80 - 3.18i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.850 + 2.61i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.76 + 9.30i)T + (-27.5 - 84.6i)T^{2} \) |
| 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
−12.01941205967468761748697198236, −11.50262085804835192189566951269, −10.35547521308850731002274861566, −9.323844675723046698814876230597, −8.029589846507068576381238745850, −6.84626180784438442727152286093, −5.88038340845562968049415419917, −4.75030624747147272493587755358, −3.99856560452456310716319253955, −2.68246140832912931308376610127,
1.05783603266756615678429766402, 3.07926216743664141754454994498, 4.33115430101213874783415324910, 5.34468235443148579058386259476, 6.60669583398475870975781784672, 7.41537849939378441280795657523, 8.057972203893138222669277307466, 9.969740451652716656695249119882, 11.01174103295459542657886618296, 11.81288194348945834932447344220
