L(s) = 1 | + (−0.839 + 1.13i)2-s + (−3.16 + 1.02i)3-s + (−0.589 − 1.91i)4-s + (0.0858 − 2.23i)5-s + (1.48 − 4.46i)6-s + (−0.838 + 0.838i)7-s + (2.66 + 0.934i)8-s + (6.51 − 4.73i)9-s + (2.47 + 1.97i)10-s + (−1.86 + 0.295i)11-s + (3.82 + 5.43i)12-s + (0.377 − 0.274i)13-s + (−0.249 − 1.65i)14-s + (2.02 + 7.15i)15-s + (−3.30 + 2.25i)16-s + (2.08 + 4.08i)17-s + ⋯ |
L(s) = 1 | + (−0.593 + 0.804i)2-s + (−1.82 + 0.593i)3-s + (−0.294 − 0.955i)4-s + (0.0384 − 0.999i)5-s + (0.606 − 1.82i)6-s + (−0.316 + 0.316i)7-s + (0.943 + 0.330i)8-s + (2.17 − 1.57i)9-s + (0.781 + 0.624i)10-s + (−0.563 + 0.0892i)11-s + (1.10 + 1.56i)12-s + (0.104 − 0.0760i)13-s + (−0.0667 − 0.443i)14-s + (0.522 + 1.84i)15-s + (−0.826 + 0.563i)16-s + (0.504 + 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.158403 + 0.312041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.158403 + 0.312041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.839 - 1.13i)T \) |
| 5 | \( 1 + (-0.0858 + 2.23i)T \) |
good | 3 | \( 1 + (3.16 - 1.02i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (0.838 - 0.838i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.86 - 0.295i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-0.377 + 0.274i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.08 - 4.08i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.912 + 1.79i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (0.627 + 3.96i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (3.25 - 6.38i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (-5.55 - 1.80i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.92 - 2.12i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.14 - 4.32i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + (5.11 - 10.0i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (11.6 - 3.79i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.89 - 1.25i)T + (56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (-6.71 + 1.06i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.330 - 1.01i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.28 - 10.1i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.93 + 1.25i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.203 - 0.625i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.57 + 1.16i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.478 + 0.347i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 5.50i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.31952165585663048344672508787, −10.56855866649949956895494495289, −9.828880151605020727178983800713, −8.997352674838842878282903330465, −7.86145048098838722432768732509, −6.54719221542531755602954443744, −5.87160583016564294098919676554, −5.08838637730485897330036535835, −4.32022002988305591867496014294, −1.10025821940614343140064126949,
0.43222391714228316238880530152, 2.16410284801520180302075317673, 3.80510716775633348970824949660, 5.22937157877928017330364635537, 6.30981353029002442944644475810, 7.25124867117704797772325981873, 7.82601552503944015374582016160, 9.773751414078950144416254710655, 10.21577476573885080382931581169, 11.14279793280051047221985337758