| L(s) = 1 | + (0.381 + 2.32i)2-s + (0.267 + 0.963i)3-s + (−3.37 + 1.13i)4-s + (−0.119 + 0.175i)5-s + (−2.14 + 0.990i)6-s + (−2.79 − 0.615i)7-s + (−1.72 − 3.25i)8-s + (−0.856 + 0.515i)9-s + (−0.454 − 0.210i)10-s + (3.07 + 2.33i)11-s + (−1.99 − 2.94i)12-s + (1.94 + 1.17i)13-s + (0.365 − 6.74i)14-s + (−0.201 − 0.0678i)15-s + (1.25 − 0.950i)16-s + (2.86 − 0.631i)17-s + ⋯ |
| L(s) = 1 | + (0.269 + 1.64i)2-s + (0.154 + 0.556i)3-s + (−1.68 + 0.568i)4-s + (−0.0532 + 0.0786i)5-s + (−0.873 + 0.404i)6-s + (−1.05 − 0.232i)7-s + (−0.610 − 1.15i)8-s + (−0.285 + 0.171i)9-s + (−0.143 − 0.0665i)10-s + (0.925 + 0.703i)11-s + (−0.577 − 0.851i)12-s + (0.540 + 0.324i)13-s + (0.0977 − 1.80i)14-s + (−0.0519 − 0.0175i)15-s + (0.312 − 0.237i)16-s + (0.695 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0431886 + 1.18707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0431886 + 1.18707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.267 - 0.963i)T \) |
| 59 | \( 1 + (-2.86 + 7.12i)T \) |
| good | 2 | \( 1 + (-0.381 - 2.32i)T + (-1.89 + 0.638i)T^{2} \) |
| 5 | \( 1 + (0.119 - 0.175i)T + (-1.85 - 4.64i)T^{2} \) |
| 7 | \( 1 + (2.79 + 0.615i)T + (6.35 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 2.33i)T + (2.94 + 10.5i)T^{2} \) |
| 13 | \( 1 + (-1.94 - 1.17i)T + (6.08 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-2.86 + 0.631i)T + (15.4 - 7.13i)T^{2} \) |
| 19 | \( 1 + (-1.52 - 0.165i)T + (18.5 + 4.08i)T^{2} \) |
| 23 | \( 1 + (-2.92 + 3.44i)T + (-3.72 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.985 + 6.01i)T + (-27.4 - 9.25i)T^{2} \) |
| 31 | \( 1 + (5.08 - 0.552i)T + (30.2 - 6.66i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 8.05i)T + (-20.7 - 30.6i)T^{2} \) |
| 41 | \( 1 + (-5.24 - 6.17i)T + (-6.63 + 40.4i)T^{2} \) |
| 43 | \( 1 + (7.23 - 5.50i)T + (11.5 - 41.4i)T^{2} \) |
| 47 | \( 1 + (-2.11 - 3.12i)T + (-17.3 + 43.6i)T^{2} \) |
| 53 | \( 1 + (1.46 - 0.678i)T + (34.3 - 40.3i)T^{2} \) |
| 61 | \( 1 + (0.141 + 0.860i)T + (-57.8 + 19.4i)T^{2} \) |
| 67 | \( 1 + (5.50 + 10.3i)T + (-37.5 + 55.4i)T^{2} \) |
| 71 | \( 1 + (-5.57 - 8.21i)T + (-26.2 + 65.9i)T^{2} \) |
| 73 | \( 1 + (-0.730 + 13.4i)T + (-72.5 - 7.89i)T^{2} \) |
| 79 | \( 1 + (-3.23 + 11.6i)T + (-67.6 - 40.7i)T^{2} \) |
| 83 | \( 1 + (8.02 - 7.60i)T + (4.49 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-1.11 + 6.79i)T + (-84.3 - 28.4i)T^{2} \) |
| 97 | \( 1 + (-0.861 - 15.8i)T + (-96.4 + 10.4i)T^{2} \) |
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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
−13.47362340945621132037877139260, −12.60675242392910007087414857686, −11.14979848602650086202330838354, −9.641446893123297381796589585830, −9.141982881087230228495748715957, −7.79699618791464558779615696729, −6.80822066723897178253544960954, −5.98803631654891214032968063097, −4.64325776327873751874672036423, −3.54728331041532441738009135500,
1.14399049172578143311099626506, 2.95521593085774100197114041252, 3.74284690966761485540580981481, 5.57390914816700637743603368281, 6.86016963272517149187076071939, 8.587418720312179455676801121974, 9.393898892728349310235840930304, 10.40675647186896074868318365431, 11.44565508395483390138881094169, 12.20948153690334768314249313078
