| L(s) = 1 | + (−1.28 − 2.14i)2-s + (−2.74 + 1.21i)3-s + (−1.04 + 1.97i)4-s + (−0.792 − 7.28i)5-s + (6.13 + 4.31i)6-s + (−2.16 + 0.729i)7-s + (−4.38 + 0.237i)8-s + (6.05 − 6.65i)9-s + (−14.5 + 11.0i)10-s + (4.14 + 2.81i)11-s + (0.478 − 6.70i)12-s + (−14.8 − 14.0i)13-s + (4.35 + 3.69i)14-s + (11.0 + 19.0i)15-s + (11.1 + 16.5i)16-s + (−5.14 + 15.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 1.07i)2-s + (−0.914 + 0.404i)3-s + (−0.262 + 0.494i)4-s + (−0.158 − 1.45i)5-s + (1.02 + 0.718i)6-s + (−0.309 + 0.104i)7-s + (−0.548 + 0.0297i)8-s + (0.672 − 0.739i)9-s + (−1.45 + 1.10i)10-s + (0.377 + 0.255i)11-s + (0.0398 − 0.558i)12-s + (−1.14 − 1.08i)13-s + (0.310 + 0.263i)14-s + (0.734 + 1.26i)15-s + (0.699 + 1.03i)16-s + (−0.302 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0120 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0120 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0689025 + 0.0697360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0689025 + 0.0697360i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.74 - 1.21i)T \) |
| 59 | \( 1 + (58.8 + 4.55i)T \) |
| good | 2 | \( 1 + (1.28 + 2.14i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (0.792 + 7.28i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (2.16 - 0.729i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-4.14 - 2.81i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (14.8 + 14.0i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (5.14 - 15.2i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-5.67 - 34.5i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (-5.80 + 1.61i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (9.42 - 15.6i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-4.77 + 29.1i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.242 + 4.47i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (21.6 + 6.01i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (44.2 + 65.2i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (5.05 - 46.4i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-15.0 + 19.7i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (21.4 - 12.8i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (5.18 + 95.6i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-6.89 + 63.4i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (25.4 - 29.9i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (30.8 - 77.3i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-17.6 - 38.2i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-52.8 + 87.8i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-58.3 - 68.7i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−11.84681768442119540348176985430, −10.57381764754033332165725741675, −9.887778004943193716961339004687, −9.108374222684008438116202142790, −7.938930633325993076420769373984, −6.06581009503117868922653093880, −5.07697348170306735585874238768, −3.70592491052217157005318576996, −1.50763515643700105585822870572, −0.082014724136932865229828938433,
2.81823432371548482746140253761, 4.93584979255058576960162106705, 6.54688652786485789168687193235, 6.83353024236335653318026377807, 7.52622201670235539934081240197, 9.145685930562934846309260016232, 10.09813790358632178828844271677, 11.42812790216278883339085983394, 11.76914964150582423146272742792, 13.34286600782454870697505507904
