| L(s) = 1 | + (1.14 − 1.29i)3-s + (0.617 + 3.49i)5-s + (0.244 + 0.205i)7-s + (−0.377 − 2.97i)9-s + (−0.773 + 4.38i)11-s + (4.39 + 1.60i)13-s + (5.25 + 3.20i)15-s + (−0.567 − 0.982i)17-s + (0.928 − 1.60i)19-s + (0.547 − 0.0828i)21-s + (−0.110 + 0.0926i)23-s + (−7.16 + 2.60i)25-s + (−4.29 − 2.91i)27-s + (4.09 − 1.49i)29-s + (−0.514 + 0.431i)31-s + ⋯ |
| L(s) = 1 | + (0.661 − 0.750i)3-s + (0.275 + 1.56i)5-s + (0.0925 + 0.0776i)7-s + (−0.125 − 0.992i)9-s + (−0.233 + 1.32i)11-s + (1.21 + 0.443i)13-s + (1.35 + 0.827i)15-s + (−0.137 − 0.238i)17-s + (0.213 − 0.369i)19-s + (0.119 − 0.0180i)21-s + (−0.0230 + 0.0193i)23-s + (−1.43 + 0.521i)25-s + (−0.827 − 0.561i)27-s + (0.760 − 0.276i)29-s + (−0.0924 + 0.0775i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78074 + 0.354216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78074 + 0.354216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.14 + 1.29i)T \) |
| good | 5 | \( 1 + (-0.617 - 3.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.244 - 0.205i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.773 - 4.38i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.39 - 1.60i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.567 + 0.982i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.928 + 1.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.110 - 0.0926i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.09 + 1.49i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.514 - 0.431i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.79 + 6.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.04 + 0.744i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.23 + 6.98i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.91 - 6.63i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 0.805T + 53T^{2} \) |
| 59 | \( 1 + (0.517 + 2.93i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.67 + 2.24i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.99 - 2.54i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.04 + 7.01i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.30 - 12.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.8 - 4.30i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.08 - 1.85i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.52 - 4.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.24 + 18.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.18795227682367882489658065832, −10.31613521627621860977838126433, −9.411389002222647721326775837801, −8.402374741581753811090933588228, −7.17263536551362363505508300761, −6.91857130381240653392646117751, −5.82167454500658914759873315693, −4.03181684052467807528320154247, −2.85999481685813448805564262019, −1.90485053444794798586844392883,
1.26959327938067227957567673431, 3.13338502996299308233611571447, 4.21744619752883957760540334581, 5.24516723444022065105605791794, 6.03693270783237225116983357486, 7.944337041004614795077222715246, 8.571527281572917869602679911121, 9.001121219576412798839790747680, 10.14465839039545575644593956236, 10.91743393018801564045587111869
