L(s) = 1 | + (−1.29 − 0.571i)2-s + (1.48 + 0.886i)3-s + (1.34 + 1.47i)4-s + (−2.64 + 3.02i)5-s + (−1.41 − 1.99i)6-s + (−4.44 + 0.585i)7-s + (−0.896 − 2.68i)8-s + (1.42 + 2.63i)9-s + (5.15 − 2.39i)10-s + (−0.255 + 0.518i)11-s + (0.691 + 3.39i)12-s + (2.16 − 6.37i)13-s + (6.08 + 1.78i)14-s + (−6.62 + 2.14i)15-s + (−0.374 + 3.98i)16-s + (−1.08 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.914 − 0.404i)2-s + (0.859 + 0.511i)3-s + (0.673 + 0.739i)4-s + (−1.18 + 1.35i)5-s + (−0.578 − 0.815i)6-s + (−1.68 + 0.221i)7-s + (−0.316 − 0.948i)8-s + (0.475 + 0.879i)9-s + (1.63 − 0.756i)10-s + (−0.0770 + 0.156i)11-s + (0.199 + 0.979i)12-s + (0.600 − 1.76i)13-s + (1.62 + 0.476i)14-s + (−1.70 + 0.554i)15-s + (−0.0936 + 0.995i)16-s + (−0.263 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0312755 - 0.168923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0312755 - 0.168923i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 + 0.571i)T \) |
| 3 | \( 1 + (-1.48 - 0.886i)T \) |
good | 5 | \( 1 + (2.64 - 3.02i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (4.44 - 0.585i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (0.255 - 0.518i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (-2.16 + 6.37i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (1.08 - 1.08i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.700 + 3.51i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (0.182 - 1.38i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (4.44 + 0.291i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (1.28 + 0.739i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.465 - 2.34i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (0.0670 - 0.509i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (-2.60 - 1.28i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (5.59 + 1.49i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.69 + 5.52i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (6.54 - 7.45i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (0.792 - 12.0i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (13.4 - 6.63i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-0.626 + 1.51i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (4.95 + 11.9i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (0.658 - 2.45i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (6.13 - 5.37i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-5.30 - 2.19i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-3.62 + 2.09i)T + (48.5 - 84.0i)T^{2} \) |
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
−10.72662469406441891883493004813, −10.43017929950608066939483248272, −9.555175743104606680218865169459, −8.643631803832584912467429051673, −7.74730209851031201672993963766, −7.13603826182777353278544128922, −6.12997611810158575435875472198, −3.92275992354411795678107989349, −3.16661906482073579156339506123, −2.76619479242865612383417339583,
0.11450221508492032748063659455, 1.61725316419805756315533051127, 3.38283192809081015463652687270, 4.33515418663912155442545691535, 6.07860138557830280101302036030, 6.89972183960626468120562750713, 7.68456727247692695844252490795, 8.600486583719646451719784122904, 9.173454550927640519275752541787, 9.649341236347437348357463170503