L(s) = 1 | + (0.314 − 1.37i)2-s + (−2.66 + 1.54i)3-s + (−1.80 − 0.867i)4-s + (−0.5 + 0.866i)5-s + (1.28 + 4.16i)6-s + (2.53 − 0.754i)7-s + (−1.76 + 2.21i)8-s + (3.24 − 5.61i)9-s + (1.03 + 0.961i)10-s + (−1.64 − 2.84i)11-s + (6.14 − 0.459i)12-s + 6.72·13-s + (−0.241 − 3.73i)14-s − 3.08i·15-s + (2.49 + 3.12i)16-s + (3.32 − 1.91i)17-s + ⋯ |
L(s) = 1 | + (0.222 − 0.974i)2-s + (−1.54 + 0.889i)3-s + (−0.900 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.524 + 1.69i)6-s + (0.958 − 0.285i)7-s + (−0.623 + 0.781i)8-s + (1.08 − 1.87i)9-s + (0.327 + 0.304i)10-s + (−0.495 − 0.857i)11-s + (1.77 − 0.132i)12-s + 1.86·13-s + (−0.0645 − 0.997i)14-s − 0.795i·15-s + (0.623 + 0.781i)16-s + (0.806 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.739812 - 0.420942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.739812 - 0.420942i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.314 + 1.37i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.53 + 0.754i)T \) |
good | 3 | \( 1 + (2.66 - 1.54i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.64 + 2.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 + (-3.32 + 1.91i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.618 + 0.357i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 0.631i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.04iT - 29T^{2} \) |
| 31 | \( 1 + (-0.0335 - 0.0581i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.498 + 0.287i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.230iT - 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + (4.23 - 7.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.16 + 1.25i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.986 + 0.569i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0888 - 0.153i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 12.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.0iT - 71T^{2} \) |
| 73 | \( 1 + (3.89 - 2.25i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.66 - 5.58i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.24iT - 83T^{2} \) |
| 89 | \( 1 + (13.3 + 7.68i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.76iT - 97T^{2} \) |
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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
−11.32363648694394231131039651551, −10.96257154357603390372769614584, −10.43320021716444051652939088313, −9.225148031369968756176620504385, −8.018642790998201682262839269566, −6.17236286136351737640582703812, −5.45499069355111914560049664036, −4.40139497572390530195113529973, −3.44128681027363936189780655671, −0.949549166586966896043797722723,
1.28284863937573339702463316464, 4.24117192629289439927953729114, 5.32516598425902442881756481041, 5.89631164662796711275605412493, 7.00759385445740135983460282098, 7.87709325425899044762746533403, 8.689981749213259155728188883829, 10.37452246599878467577566616845, 11.30792225396996279882440620233, 12.20790467060897527467789337669