| L(s) = 1 | + (3.13 + 0.341i)2-s + (1.94 − 2.28i)3-s + (1.90 + 0.419i)4-s + (−14.9 − 11.3i)5-s + (6.87 − 6.50i)6-s + (−10.7 + 26.9i)7-s + (−18.0 − 6.09i)8-s + (−1.45 − 8.88i)9-s + (−42.9 − 40.6i)10-s + (−8.96 + 4.14i)11-s + (4.66 − 3.54i)12-s + (12.5 − 76.7i)13-s + (−42.8 + 80.8i)14-s + (−54.9 + 12.0i)15-s + (−68.8 − 31.8i)16-s + (21.7 + 54.4i)17-s + ⋯ |
| L(s) = 1 | + (1.10 + 0.120i)2-s + (0.373 − 0.440i)3-s + (0.238 + 0.0524i)4-s + (−1.33 − 1.01i)5-s + (0.467 − 0.442i)6-s + (−0.579 + 1.45i)7-s + (−0.799 − 0.269i)8-s + (−0.0539 − 0.328i)9-s + (−1.35 − 1.28i)10-s + (−0.245 + 0.113i)11-s + (0.112 − 0.0852i)12-s + (0.268 − 1.63i)13-s + (−0.818 + 1.54i)14-s + (−0.945 + 0.208i)15-s + (−1.07 − 0.497i)16-s + (0.309 + 0.777i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.139497 - 0.867828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139497 - 0.867828i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-97.7 + 442. i)T \) |
| good | 2 | \( 1 + (-3.13 - 0.341i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (14.9 + 11.3i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (10.7 - 26.9i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (8.96 - 4.14i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (-12.5 + 76.7i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-21.7 - 54.4i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (65.6 + 96.8i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (5.77 - 106. i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (184. - 20.0i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (-170. + 251. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-204. + 68.8i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (-1.98 - 36.5i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (81.8 + 37.8i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (-119. + 90.6i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (134. - 127. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (308. + 33.5i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-276. - 93.3i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (164. - 125. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (-219. + 414. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (665. + 783. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (738. - 444. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (1.03e3 - 112. i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (-711. - 1.34e3i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−12.23913044856953264069891020212, −11.38384441149140583079495093656, −9.442761886038628969062240587582, −8.569445842400132898480262191189, −7.76811206785113757857381309685, −6.06735439539070438340082717286, −5.24174927176168415327603859941, −3.93281336538768438782423399251, −2.83676974852343276236190489448, −0.25762207788527726839136098150,
2.99956032638976132889647317832, 3.95762577513547403112395471443, 4.38838035022350333138878682421, 6.43303478589593673162543268966, 7.27672025681483682081208274833, 8.497099672854082558109824848749, 9.942515531663472407921037118258, 10.92934862504943155505512995614, 11.70791725508424584287115096321, 12.76568841337099641472000510208
