| L(s) = 1 | + (−0.786 − 0.618i)2-s + (0.396 − 0.556i)3-s + (0.235 + 0.971i)4-s + (−1.47 + 0.284i)5-s + (−0.655 + 0.192i)6-s + (−2.47 + 0.947i)7-s + (0.415 − 0.909i)8-s + (0.828 + 2.39i)9-s + (1.33 + 0.687i)10-s + (−2.54 + 2.00i)11-s + (0.634 + 0.254i)12-s + (−0.626 + 0.402i)13-s + (2.52 + 0.782i)14-s + (−0.426 + 0.933i)15-s + (−0.888 + 0.458i)16-s + (1.81 + 1.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (0.228 − 0.321i)3-s + (0.117 + 0.485i)4-s + (−0.659 + 0.127i)5-s + (−0.267 + 0.0786i)6-s + (−0.933 + 0.357i)7-s + (0.146 − 0.321i)8-s + (0.276 + 0.797i)9-s + (0.421 + 0.217i)10-s + (−0.767 + 0.603i)11-s + (0.183 + 0.0733i)12-s + (−0.173 + 0.111i)13-s + (0.675 + 0.209i)14-s + (−0.110 + 0.240i)15-s + (−0.222 + 0.114i)16-s + (0.440 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0160 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0160 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.354229 + 0.359966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.354229 + 0.359966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (2.47 - 0.947i)T \) |
| 23 | \( 1 + (-3.38 + 3.39i)T \) |
| good | 3 | \( 1 + (-0.396 + 0.556i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (1.47 - 0.284i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (2.54 - 2.00i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (0.626 - 0.402i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.81 - 1.73i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.40 - 4.20i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-3.07 + 0.903i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (6.34 - 0.606i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-1.21 - 3.49i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (3.03 + 3.50i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.02 - 11.0i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (0.420 - 0.727i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0725 + 1.52i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (9.67 + 4.98i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-2.98 - 4.19i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (8.65 - 3.46i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.859 + 5.98i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.87 + 11.8i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.505 + 10.6i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-5.52 + 6.37i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.47 + 0.332i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-5.59 - 6.45i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.94528293670455278397410163375, −10.67790168307872126287885330013, −10.18585846272053963084015613759, −9.044456645252465052791314173024, −8.005210085271225580307515833327, −7.41534133580543933060890458587, −6.20979628843307912297297493072, −4.62032461387298975796616262367, −3.25910409955574143201812407872, −2.05486216552230183070404179363,
0.40370869326184352545592666348, 2.99437143619130497834999084327, 4.13140655959650704640505469224, 5.56556238189391961830042993195, 6.76767948964398263104670123557, 7.53097914068245848611916082193, 8.681372114226020121028554995469, 9.393418503125148587503935015132, 10.30939647595500986131237872008, 11.15318018948064025785164284166
