| L(s) = 1 | + (0.896 − 0.326i)2-s + (−0.0971 + 0.266i)3-s + (−0.834 + 0.700i)4-s + (1.37 + 1.76i)5-s + 0.270i·6-s + (0.0783 − 0.0138i)7-s + (−1.47 + 2.55i)8-s + (2.23 + 1.87i)9-s + (1.80 + 1.13i)10-s + (2.24 − 3.87i)11-s + (−0.105 − 0.290i)12-s + (−0.642 + 0.539i)13-s + (0.0657 − 0.0379i)14-s + (−0.604 + 0.194i)15-s + (−0.109 + 0.622i)16-s + (−3.63 − 3.04i)17-s + ⋯ |
| L(s) = 1 | + (0.633 − 0.230i)2-s + (−0.0560 + 0.154i)3-s + (−0.417 + 0.350i)4-s + (0.613 + 0.789i)5-s + 0.110i·6-s + (0.0296 − 0.00522i)7-s + (−0.521 + 0.902i)8-s + (0.745 + 0.625i)9-s + (0.570 + 0.359i)10-s + (0.675 − 1.16i)11-s + (−0.0305 − 0.0839i)12-s + (−0.178 + 0.149i)13-s + (0.0175 − 0.0101i)14-s + (−0.156 + 0.0502i)15-s + (−0.0274 + 0.155i)16-s + (−0.881 − 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46477 + 0.468518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46477 + 0.468518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.37 - 1.76i)T \) |
| 37 | \( 1 + (-4.06 + 4.52i)T \) |
| good | 2 | \( 1 + (-0.896 + 0.326i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (0.0971 - 0.266i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-0.0783 + 0.0138i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.24 + 3.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.642 - 0.539i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.63 + 3.04i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.698 + 1.92i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (0.00227 + 0.00393i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.499 + 0.288i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.06iT - 31T^{2} \) |
| 41 | \( 1 + (4.55 - 3.81i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 + (-6.88 + 3.97i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.788 - 0.138i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.84 - 1.38i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.86 + 3.40i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.19 - 0.563i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.63 - 2.05i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 9.38iT - 73T^{2} \) |
| 79 | \( 1 + (6.85 - 1.20i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (9.01 - 10.7i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.5 - 1.86i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (3.54 + 6.13i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−13.02362076317236115493718193249, −11.59160047926604234372113211781, −11.03202542137684098564340020958, −9.737763015011039730022590040475, −8.831203491693590693525625193959, −7.47326267977075321823996157482, −6.24928923705735242091645158968, −5.03148795087019742033540829463, −3.82242696198651886881871804873, −2.49227221124207929754449047756,
1.51875399280214174363190444803, 4.02921013892395624162587273089, 4.85327241108053027683478890513, 6.09668271110738208121661520591, 6.97925465825021301714050724132, 8.677174013278285412746560694772, 9.581608709570221625592414979212, 10.22002898289315025921768510719, 12.03985352669580712180821827650, 12.69802232329644579115070818025
