| L(s) = 1 | + (0.968 + 0.812i)3-s + (−0.681 − 1.87i)5-s + (−2.99 + 1.09i)7-s + (−0.243 − 1.37i)9-s + (−2.73 − 4.73i)11-s + (−1.50 − 0.265i)13-s + (0.861 − 2.36i)15-s + (−0.794 + 0.140i)17-s + (1.35 − 1.61i)19-s + (−3.79 − 1.38i)21-s + (−3.02 − 1.74i)23-s + (0.788 − 0.661i)25-s + (2.78 − 4.81i)27-s + (−0.122 + 0.0705i)29-s + 3.37i·31-s + ⋯ |
| L(s) = 1 | + (0.559 + 0.469i)3-s + (−0.304 − 0.837i)5-s + (−1.13 + 0.412i)7-s + (−0.0810 − 0.459i)9-s + (−0.823 − 1.42i)11-s + (−0.418 − 0.0737i)13-s + (0.222 − 0.611i)15-s + (−0.192 + 0.0339i)17-s + (0.311 − 0.371i)19-s + (−0.827 − 0.301i)21-s + (−0.631 − 0.364i)23-s + (0.157 − 0.132i)25-s + (0.535 − 0.927i)27-s + (−0.0226 + 0.0131i)29-s + 0.606i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.478601 - 0.725474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.478601 - 0.725474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (-0.539 - 6.05i)T \) |
| good | 3 | \( 1 + (-0.968 - 0.812i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.681 + 1.87i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.99 - 1.09i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.50 + 0.265i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.794 - 0.140i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-1.35 + 1.61i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (3.02 + 1.74i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.122 - 0.0705i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.37iT - 31T^{2} \) |
| 41 | \( 1 + (-0.0732 + 0.415i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 2.00iT - 43T^{2} \) |
| 47 | \( 1 + (-0.842 + 1.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.86 + 2.49i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-3.65 + 10.0i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 2.00i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.25 + 2.64i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.78 + 5.69i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 8.77T + 73T^{2} \) |
| 79 | \( 1 + (1.54 + 4.24i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.81 - 10.3i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-0.914 + 2.51i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (13.3 + 7.71i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.15247288988688355026721855096, −9.478793388516893263930006597977, −8.620216682178939907951125830750, −8.191542551083583806849542214777, −6.70206869819667359066257103814, −5.79663433959305207926446703519, −4.73358994472644872835491156623, −3.48507449272731708644201393359, −2.78926905763320187184655559498, −0.42633180450983719234184311879,
2.15478877418299848872591962136, 3.01639293612934220446786823366, 4.19292878091366625357148999536, 5.53109853300670697038237758719, 6.88795095633988179168229695260, 7.30761556698051942548385096896, 8.026890152729561711543432256632, 9.382444254971652895966373337288, 10.09259255688774042067013384534, 10.74777924798059001013017397843
