L(s) = 1 | + (−0.890 − 1.09i)2-s + (−0.706 + 1.58i)3-s + (−0.412 + 1.95i)4-s + (−1.50 − 2.24i)5-s + (2.36 − 0.632i)6-s + (−0.0860 + 0.207i)7-s + (2.51 − 1.29i)8-s + (−2.00 − 2.23i)9-s + (−1.12 + 3.64i)10-s + (−3.68 − 0.732i)11-s + (−2.80 − 2.03i)12-s + (−1.53 − 1.02i)13-s + (0.304 − 0.0905i)14-s + (4.61 − 0.786i)15-s + (−3.65 − 1.61i)16-s + (−4.44 − 4.44i)17-s + ⋯ |
L(s) = 1 | + (−0.629 − 0.776i)2-s + (−0.407 + 0.913i)3-s + (−0.206 + 0.978i)4-s + (−0.671 − 1.00i)5-s + (0.966 − 0.258i)6-s + (−0.0325 + 0.0784i)7-s + (0.889 − 0.456i)8-s + (−0.667 − 0.744i)9-s + (−0.357 + 1.15i)10-s + (−1.11 − 0.220i)11-s + (−0.809 − 0.587i)12-s + (−0.425 − 0.284i)13-s + (0.0814 − 0.0241i)14-s + (1.19 − 0.203i)15-s + (−0.914 − 0.403i)16-s + (−1.07 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0250747 - 0.214460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0250747 - 0.214460i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.890 + 1.09i)T \) |
| 3 | \( 1 + (0.706 - 1.58i)T \) |
good | 5 | \( 1 + (1.50 + 2.24i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.0860 - 0.207i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (3.68 + 0.732i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.53 + 1.02i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (4.44 + 4.44i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.66 + 2.45i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.88 - 4.55i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 7.06i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 + (5.00 + 7.48i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (4.55 - 1.88i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.54 + 0.307i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-4.26 + 4.26i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.02 + 5.13i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (7.47 - 4.99i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-0.815 - 4.10i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (8.05 - 1.60i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (2.52 + 1.04i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.90 + 2.44i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (8.79 - 8.79i)T - 79iT^{2} \) |
| 83 | \( 1 + (-4.93 + 7.39i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (5.44 + 2.25i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 16.3iT - 97T^{2} \) |
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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
−11.89055340975023631444061061524, −11.02829895238887624959338505048, −10.24023815507320519502970201473, −9.045250093697862099699704687029, −8.551969183134783222862379494783, −7.21337380455634352954950177316, −5.15804814752536256079417404517, −4.40442507516008734045864364101, −2.90828926601366932020056242272, −0.23226156912467906521001749464,
2.32681323039303958885338750722, 4.64847231265795395719913951492, 6.20645845484324922517170372890, 6.83295406289217811464806469227, 7.83386497530344149601316749703, 8.497720665794850772904675541027, 10.34363907192515225074113343237, 10.75286912530003536749563414214, 11.91472229354009700966733146326, 13.09917349198933536146872533741