| L(s) = 1 | + (1.00 − 1.11i)2-s + (0.317 + 3.02i)3-s + (−0.0280 − 0.267i)4-s + (1.28 − 1.82i)5-s + (3.70 + 2.68i)6-s + (−2.41 + 1.09i)7-s + (2.10 + 1.53i)8-s + (−6.08 + 1.29i)9-s + (−0.751 − 3.28i)10-s + (−2.05 − 0.436i)11-s + (0.797 − 0.169i)12-s + (1.23 − 3.79i)13-s + (−1.20 + 3.79i)14-s + (5.93 + 3.30i)15-s + (4.36 − 0.928i)16-s + (6.53 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (0.712 − 0.791i)2-s + (0.183 + 1.74i)3-s + (−0.0140 − 0.133i)4-s + (0.574 − 0.818i)5-s + (1.51 + 1.09i)6-s + (−0.911 + 0.412i)7-s + (0.745 + 0.541i)8-s + (−2.02 + 0.431i)9-s + (−0.237 − 1.03i)10-s + (−0.619 − 0.131i)11-s + (0.230 − 0.0489i)12-s + (0.341 − 1.05i)13-s + (−0.323 + 1.01i)14-s + (1.53 + 0.852i)15-s + (1.09 − 0.232i)16-s + (1.58 − 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68509 + 0.390757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68509 + 0.390757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.28 + 1.82i)T \) |
| 7 | \( 1 + (2.41 - 1.09i)T \) |
| good | 2 | \( 1 + (-1.00 + 1.11i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.317 - 3.02i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (2.05 + 0.436i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 3.79i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.53 + 2.90i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.355 + 3.38i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.89 - 2.10i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (4.73 - 3.44i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.40 - 1.51i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.0794 - 0.0168i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (0.937 - 2.88i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + (1.47 + 0.656i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (0.668 + 6.35i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (1.05 + 1.17i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (6.94 - 7.70i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.92 + 1.30i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (7.13 - 5.18i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-10.2 - 2.18i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.12 - 0.501i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 7.67i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.0891 - 0.0990i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-2.77 + 2.01i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.83108090928086301582041861923, −11.82626787616207987861896102385, −10.67401630240697373596057701290, −9.946138710478573803737503226311, −9.175579193973869814046997028889, −8.024657661499549205421975500425, −5.49239133039765770030026624431, −5.14910059786868155035632215983, −3.64085096549440351576420347189, −2.88172958551423887475979942345,
1.78486876037586169222641091770, 3.51460657178545347602753398102, 5.81953122738290894761626472374, 6.26931607435175976455659147609, 7.20099307469751622697816196809, 7.85600608222589638502677237268, 9.659784226920546210149165884240, 10.69134867094499548504571790445, 12.18780802597543196292719616282, 12.97923815028083943701469846181
