| L(s) = 1 | + (−15.9 + 1.16i)2-s + (−140. − 58.0i)3-s + (253. − 37.1i)4-s + (63.8 − 26.4i)5-s + (2.30e3 + 763. i)6-s + (1.17e3 + 1.17e3i)7-s + (−3.99e3 + 887. i)8-s + (1.16e4 + 1.16e4i)9-s + (−988. + 496. i)10-s + (1.60e4 − 6.64e3i)11-s + (−3.76e4 − 9.50e3i)12-s + (−2.65e4 − 1.09e4i)13-s + (−2.01e4 − 1.73e4i)14-s − 1.04e4·15-s + (6.27e4 − 1.88e4i)16-s + 4.21e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0726i)2-s + (−1.73 − 0.716i)3-s + (0.989 − 0.145i)4-s + (0.102 − 0.0423i)5-s + (1.77 + 0.589i)6-s + (0.489 + 0.489i)7-s + (−0.976 + 0.216i)8-s + (1.77 + 1.77i)9-s + (−0.0988 + 0.0496i)10-s + (1.09 − 0.454i)11-s + (−1.81 − 0.458i)12-s + (−0.928 − 0.384i)13-s + (−0.523 − 0.452i)14-s − 0.207·15-s + (0.957 − 0.286i)16-s + 0.504i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.189763 - 0.367698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189763 - 0.367698i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.9 - 1.16i)T \) |
| good | 3 | \( 1 + (140. + 58.0i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-63.8 + 26.4i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-1.17e3 - 1.17e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (-1.60e4 + 6.64e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (2.65e4 + 1.09e4i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 4.21e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-5.35e4 + 1.29e5i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-5.13e3 + 5.13e3i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (4.07e5 - 9.84e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 1.52e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.57e6 + 1.06e6i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-2.62e5 - 2.62e5i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (1.45e6 - 6.04e5i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 2.42e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (3.18e6 + 7.67e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (7.48e6 + 1.80e7i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-4.91e6 + 1.18e7i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (5.72e6 + 2.37e6i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-4.34e5 - 4.34e5i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (2.97e7 + 2.97e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 1.20e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (9.43e6 - 2.27e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (5.71e7 - 5.71e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 - 1.66e8T + 7.83e15T^{2} \) |
| 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
−14.94754607015246324273390035364, −12.82832970051146963855099052377, −11.63946850100776931474564110630, −11.12756076973301488457656033693, −9.538434268220317051696050481727, −7.70010990049352572305555308460, −6.48841958120718313713499462736, −5.35167507951142364636748487836, −1.71904990207070438477103156888, −0.35154973689364990962063070632,
1.21911214825784006619304904788, 4.35963436413347103154813660938, 6.06268534155073681520522482966, 7.30058374954158981834364215433, 9.539619262609260723855570732855, 10.29646448899271386895497553472, 11.59861131603004959387509453236, 12.08834066558431205186966253715, 14.66216723317098472367522996573, 15.98984050284767869748995243107
