| L(s) = 1 | + (−2.32 + 0.350i)2-s + (3.37 − 1.04i)4-s + (−0.0416 + 0.555i)5-s + (−1.55 + 2.13i)7-s + (−3.24 + 1.56i)8-s + (−0.0979 − 1.30i)10-s + (−0.266 + 0.678i)11-s + (−1.59 − 2.00i)13-s + (2.86 − 5.52i)14-s + (1.16 − 0.792i)16-s + (1.02 − 0.954i)17-s + (−2.19 + 3.80i)19-s + (0.437 + 1.91i)20-s + (0.381 − 1.67i)22-s + (−3.73 − 3.46i)23-s + ⋯ |
| L(s) = 1 | + (−1.64 + 0.247i)2-s + (1.68 − 0.520i)4-s + (−0.0186 + 0.248i)5-s + (−0.588 + 0.808i)7-s + (−1.14 + 0.552i)8-s + (−0.0309 − 0.413i)10-s + (−0.0802 + 0.204i)11-s + (−0.443 − 0.556i)13-s + (0.766 − 1.47i)14-s + (0.290 − 0.198i)16-s + (0.249 − 0.231i)17-s + (−0.503 + 0.872i)19-s + (0.0978 + 0.428i)20-s + (0.0812 − 0.356i)22-s + (−0.778 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00504534 - 0.149766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00504534 - 0.149766i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.55 - 2.13i)T \) |
| good | 2 | \( 1 + (2.32 - 0.350i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (0.0416 - 0.555i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.266 - 0.678i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (1.59 + 2.00i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.02 + 0.954i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.19 - 3.80i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.73 + 3.46i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.72 - 7.55i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (1.93 + 3.35i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.82 + 2.10i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (9.91 - 4.77i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (9.13 + 4.39i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (11.5 - 1.73i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (3.25 - 1.00i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.777 + 10.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-9.94 - 3.06i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.11 - 5.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.99 - 8.73i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (10.3 + 1.55i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (3.15 - 5.47i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.50 + 1.89i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-0.183 - 0.467i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
| 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
−11.27730956110568449390202456051, −10.11903337445467943479733429264, −9.981099288327131492758831511702, −8.705195919649256962067295672665, −8.273196028512873843965240676844, −7.08463931655606577830879005201, −6.41380605327621056120235685301, −5.18594850006614609711112374579, −3.20959049166009833557761002265, −1.88981890187416353948388094978,
0.15367823230026402832123461282, 1.74235217627772522257040281722, 3.26644167780391452976094433327, 4.76631816570825740060994700896, 6.49371941273051634394639203671, 7.13887064274220843089393328260, 8.167329105213649480298125984010, 8.887253222745856853623895635188, 9.894624070374125966436375607939, 10.26828435814671623216119536656
