| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−7.28 + 5.29i)3-s + (−12.9 + 9.40i)4-s + (−5.54 − 55.6i)5-s + (29.1 + 21.1i)6-s − 223.·7-s + (51.7 + 37.6i)8-s + (25.0 − 77.0i)9-s + (−204. + 89.8i)10-s + (−72.9 − 224. i)11-s + (44.4 − 136. i)12-s + (31.4 − 96.7i)13-s + (275. + 848. i)14-s + (334. + 375. i)15-s + (79.1 − 243. i)16-s + (−174. − 126. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.0992 − 0.995i)5-s + (0.330 + 0.239i)6-s − 1.72·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.647 + 0.284i)10-s + (−0.181 − 0.559i)11-s + (0.0892 − 0.274i)12-s + (0.0515 − 0.158i)13-s + (0.376 + 1.15i)14-s + (0.384 + 0.431i)15-s + (0.0772 − 0.237i)16-s + (−0.146 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4244548038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4244548038\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 + (7.28 - 5.29i)T \) |
| 5 | \( 1 + (5.54 + 55.6i)T \) |
| good | 7 | \( 1 + 223.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (72.9 + 224. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-31.4 + 96.7i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (174. + 126. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.64e3 + 1.19e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-1.22e3 - 3.76e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.24e3 + 2.35e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.88e3 - 2.82e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.19e3 - 9.82e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (729. - 2.24e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.94e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.05e3 - 3.67e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.17e4 + 1.57e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.10e3 - 9.56e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.51e4 + 4.65e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-5.02e4 - 3.65e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (3.08e4 - 2.24e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.14e3 - 9.68e3i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.93e4 - 5.76e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (3.20e4 + 2.32e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.84e4 - 8.75e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (4.28e4 - 3.11e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.21846736079366827898104577463, −11.20711265123068829261097043503, −10.05462929453686087055412567619, −9.349742539947171676945839492683, −8.409916367886740081327876891987, −6.71176267474274027356075913181, −5.49574416082671822300978923592, −4.16234665528931911498300647715, −2.94609814061400069866345727278, −0.836271972223211546687587138674,
0.23079167416464261254757892103, 2.57045418563704656093119402554, 4.13674669375250820325296655200, 6.00271298313598064200498062872, 6.58837802786579790226687363304, 7.40000210972793860197459292449, 8.866318351801009963953076619342, 10.13446190308132711178663813766, 10.63866711866721063422880753376, 12.27042350692253247009954539376
