| L(s) = 1 | + (0.212 + 0.654i)2-s + (1.65 + 0.521i)3-s + (1.23 − 0.897i)4-s + (−0.951 − 0.309i)5-s + (0.00984 + 1.19i)6-s + (−0.891 − 1.22i)7-s + (1.96 + 1.42i)8-s + (2.45 + 1.72i)9-s − 0.687i·10-s + (−2.86 − 1.66i)11-s + (2.50 − 0.837i)12-s + (−3.46 + 1.12i)13-s + (0.613 − 0.844i)14-s + (−1.40 − 1.00i)15-s + (0.427 − 1.31i)16-s + (−1.34 + 4.15i)17-s + ⋯ |
| L(s) = 1 | + (0.150 + 0.462i)2-s + (0.953 + 0.301i)3-s + (0.617 − 0.448i)4-s + (−0.425 − 0.138i)5-s + (0.00401 + 0.486i)6-s + (−0.336 − 0.463i)7-s + (0.693 + 0.504i)8-s + (0.818 + 0.574i)9-s − 0.217i·10-s + (−0.865 − 0.501i)11-s + (0.724 − 0.241i)12-s + (−0.960 + 0.312i)13-s + (0.163 − 0.225i)14-s + (−0.363 − 0.259i)15-s + (0.106 − 0.329i)16-s + (−0.327 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59933 + 0.363893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59933 + 0.363893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.521i)T \) |
| 5 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 + (2.86 + 1.66i)T \) |
| good | 2 | \( 1 + (-0.212 - 0.654i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (0.891 + 1.22i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (3.46 - 1.12i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.34 - 4.15i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.51 - 2.08i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 0.708iT - 23T^{2} \) |
| 29 | \( 1 + (-4.12 + 2.99i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.57 + 4.83i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.50 - 3.99i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.19 + 4.50i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.31iT - 43T^{2} \) |
| 47 | \( 1 + (-3.44 + 4.74i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.75 + 2.19i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.19 - 9.89i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (5.99 + 1.94i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + (10.9 + 3.54i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.87 - 5.33i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-12.6 + 4.12i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.58 - 14.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 5.44iT - 89T^{2} \) |
| 97 | \( 1 + (-3.77 - 11.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.22002930383824918038473698509, −11.95078242530140176423621948446, −10.56443469679204255549575645725, −10.08528726536418775778844969000, −8.566493524780572532539897533262, −7.69982783939204276472084344863, −6.75004377735930794146974297632, −5.24523533974492883427795964923, −3.89406308119054878402616189501, −2.28914389816609940866131300304,
2.36950044547623071493660855997, 3.14775520171032658038683122166, 4.74281545189047644247113728120, 6.85336298426611479787232794720, 7.47690179068782293578827585449, 8.568971060789982232574368638311, 9.793476791557835251369839802038, 10.79188365901108007340266177712, 12.13572441378957090519735787325, 12.56269119000788757230867832476
