L(s) = 1 | + (0.349 − 2.20i)2-s + (−1.26 − 0.642i)3-s + (−2.85 − 0.927i)4-s + (2.20 + 0.366i)5-s + (−1.85 + 2.55i)6-s + (−1.01 + 1.99i)8-s + (−0.587 − 0.809i)9-s + (1.58 − 4.74i)10-s + (3.00 + 3.00i)12-s + (−4.41 − 0.699i)13-s + (−2.54 − 1.87i)15-s + (−0.809 − 0.587i)16-s + (−4.41 + 0.699i)17-s + (−1.99 + 1.01i)18-s + (−1.95 − 6.01i)19-s + (−5.95 − 3.09i)20-s + ⋯ |
L(s) = 1 | + (0.247 − 1.56i)2-s + (−0.727 − 0.370i)3-s + (−1.42 − 0.463i)4-s + (0.986 + 0.163i)5-s + (−0.758 + 1.04i)6-s + (−0.358 + 0.704i)8-s + (−0.195 − 0.269i)9-s + (0.499 − 1.50i)10-s + (0.866 + 0.866i)12-s + (−1.22 − 0.194i)13-s + (−0.656 − 0.484i)15-s + (−0.202 − 0.146i)16-s + (−1.07 + 0.169i)17-s + (−0.469 + 0.239i)18-s + (−0.448 − 1.37i)19-s + (−1.33 − 0.691i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.405618 + 0.848878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.405618 + 0.848878i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.20 - 0.366i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.349 + 2.20i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (1.26 + 0.642i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (4.41 + 0.699i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (4.41 - 0.699i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.95 + 6.01i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1 - i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.95 + 6.01i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.78 + 1.92i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-6.01 + 1.95i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (1.92 - 3.78i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.221 + 1.39i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (5.70 + 1.85i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 5.11i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 + (-6.47 - 4.70i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.98 - 2.03i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (5.11 - 3.71i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.39 + 8.83i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + (-9.77 - 1.54i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.31290567948970536721611297330, −9.542513814632773735022065993592, −8.870880966509569672534348885370, −7.16761093626725325367720217510, −6.32957357324997518612027724993, −5.25739039116070913365979211712, −4.36202615126042498247175804392, −2.81836615160303873764005431408, −2.07845120106887990414662762857, −0.49654949681548063256370216568,
2.27942989526277649049319842440, 4.40277114423705436529110060647, 5.03946958230128504326206239585, 5.86239692107538045936573345869, 6.46484028899394460149247961094, 7.45750028527755110145875512077, 8.440847743584976785907157273340, 9.315378670777033210041275948700, 10.23183869956558133436592133479, 11.03123577351796664976946708150