| L(s) = 1 | + (5.10 + 0.554i)2-s + (−1.94 + 2.28i)3-s + (17.8 + 3.93i)4-s + (10.4 + 7.93i)5-s + (−11.1 + 10.5i)6-s + (11.7 − 29.4i)7-s + (50.2 + 16.9i)8-s + (−1.45 − 8.88i)9-s + (48.8 + 46.2i)10-s + (−44.4 + 20.5i)11-s + (−43.7 + 33.2i)12-s + (−5.27 + 32.2i)13-s + (76.1 − 143. i)14-s + (−38.4 + 8.45i)15-s + (113. + 52.6i)16-s + (−30.1 − 75.7i)17-s + ⋯ |
| L(s) = 1 | + (1.80 + 0.196i)2-s + (−0.373 + 0.440i)3-s + (2.23 + 0.492i)4-s + (0.933 + 0.709i)5-s + (−0.760 + 0.720i)6-s + (0.633 − 1.58i)7-s + (2.21 + 0.747i)8-s + (−0.0539 − 0.328i)9-s + (1.54 + 1.46i)10-s + (−1.21 + 0.564i)11-s + (−1.05 + 0.800i)12-s + (−0.112 + 0.687i)13-s + (1.45 − 2.74i)14-s + (−0.661 + 0.145i)15-s + (1.77 + 0.821i)16-s + (−0.430 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.63051 + 1.65966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.63051 + 1.65966i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.94 - 2.28i)T \) |
| 59 | \( 1 + (68.6 + 447. i)T \) |
| good | 2 | \( 1 + (-5.10 - 0.554i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (-10.4 - 7.93i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (-11.7 + 29.4i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (44.4 - 20.5i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (5.27 - 32.2i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (30.1 + 75.7i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (-61.3 - 90.4i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (-1.97 + 36.4i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (187. - 20.3i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (51.0 - 75.2i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (43.3 - 14.6i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (22.6 + 417. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (-193. - 89.2i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (24.1 - 18.3i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (-287. + 272. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (131. + 14.3i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-127. - 42.8i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (334. - 254. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (404. - 763. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-893. - 1.05e3i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (-1.08e3 + 652. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (1.23e3 - 133. i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (-209. - 394. i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−12.59752223921608800824982332126, −11.37887521243895639891721774740, −10.67791942210771744812588236322, −9.891113413375664764062817939492, −7.46891357020020146830767180984, −6.89920363578694542062420466370, −5.58259827050682759474717801685, −4.76212411720123028239982244440, −3.67777343297689803913220785040, −2.16561805428410823140085302803,
1.84568283208679104257515214192, 2.82017115750137688660611093244, 4.90109256820919215577823055463, 5.57317335906810495390239826270, 5.99015180768798029290230988662, 7.75097718658444936836992148862, 9.067683207649434829877562949473, 10.71037645775519957942310876500, 11.52835758707046350085245966509, 12.44249304498336140831260588219
