| L(s) = 1 | + (−1.47 + 2.89i)2-s + (0.270 − 1.71i)3-s + (−3.87 − 5.33i)4-s + (−3.93 − 3.08i)5-s + (4.55 + 3.31i)6-s + (−8.41 − 8.41i)7-s + (8.32 − 1.31i)8-s + (−2.85 − 0.927i)9-s + (14.7 − 6.84i)10-s + (4.07 + 12.5i)11-s + (−10.1 + 5.18i)12-s + (−2.42 − 4.75i)13-s + (36.8 − 11.9i)14-s + (−6.34 + 5.89i)15-s + (−0.328 + 1.01i)16-s + (0.874 + 5.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.738 + 1.44i)2-s + (0.0903 − 0.570i)3-s + (−0.968 − 1.33i)4-s + (−0.786 − 0.617i)5-s + (0.759 + 0.552i)6-s + (−1.20 − 1.20i)7-s + (1.04 − 0.164i)8-s + (−0.317 − 0.103i)9-s + (1.47 − 0.684i)10-s + (0.370 + 1.14i)11-s + (−0.847 + 0.431i)12-s + (−0.186 − 0.365i)13-s + (2.63 − 0.854i)14-s + (−0.422 + 0.392i)15-s + (−0.0205 + 0.0632i)16-s + (0.0514 + 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.238240 - 0.205708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238240 - 0.205708i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.270 + 1.71i)T \) |
| 5 | \( 1 + (3.93 + 3.08i)T \) |
| good | 2 | \( 1 + (1.47 - 2.89i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (8.41 + 8.41i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.07 - 12.5i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (2.42 + 4.75i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-0.874 - 5.52i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-2.03 + 2.79i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (34.7 + 17.6i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (20.4 + 28.1i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-6.74 - 4.90i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-43.7 + 22.2i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-20.8 + 64.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (2.42 - 2.42i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (20.5 + 3.25i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (6.91 - 43.6i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-10.2 - 3.33i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-10.8 - 33.2i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (14.1 + 89.4i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (25.0 - 18.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (9.90 + 5.04i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (69.0 + 95.0i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (22.3 - 3.53i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-25.3 + 8.23i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-169. - 26.9i)T + (8.94e3 + 2.90e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.32306736429024442023672174570, −13.09976371468497615075779079133, −12.15567796895084620508779355753, −10.19363826431354174729555133041, −9.209203289864082027281950614726, −7.83055515575096436009289204059, −7.24932053185546772969440533762, −6.13311476827061173855686030478, −4.18624702991238693112283771970, −0.32936469529297595114010540277,
2.80942548588233615492076946532, 3.69889121517818387569809695347, 6.14519571260698921695958220832, 8.215563301157164075479566170362, 9.273489776194568685778013204034, 10.02285970493721950381822010248, 11.39114488959410843322288612404, 11.79048846288976487935145537520, 13.01978651716502216505016981014, 14.55166489163218887874512509191
