| L(s) = 1 | + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (−2.20 − 0.349i)5-s + (−1.10 − 0.771i)7-s + (0.965 + 0.258i)8-s + (−2.16 − 0.540i)10-s + (1.46 − 4.02i)11-s + (−0.298 − 3.41i)13-s + (−1.03 − 0.864i)14-s + (0.939 + 0.342i)16-s + (−7.43 + 1.99i)17-s + (6.12 − 3.53i)19-s + (−2.11 − 0.727i)20-s + (1.81 − 3.88i)22-s + (−2.78 − 3.97i)23-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (−0.987 − 0.156i)5-s + (−0.416 − 0.291i)7-s + (0.341 + 0.0915i)8-s + (−0.686 − 0.170i)10-s + (0.442 − 1.21i)11-s + (−0.0828 − 0.947i)13-s + (−0.275 − 0.231i)14-s + (0.234 + 0.0855i)16-s + (−1.80 + 0.482i)17-s + (1.40 − 0.811i)19-s + (−0.472 − 0.162i)20-s + (0.386 − 0.828i)22-s + (−0.580 − 0.829i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07267 - 1.09825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07267 - 1.09825i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.20 + 0.349i)T \) |
| good | 7 | \( 1 + (1.10 + 0.771i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.46 + 4.02i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.298 + 3.41i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (7.43 - 1.99i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.12 + 3.53i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.78 + 3.97i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.00478 - 0.00401i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.582 + 3.30i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.24 + 4.65i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.705 - 0.841i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.14 - 6.74i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.04 + 2.92i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-8.13 + 8.13i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.71 - 3.17i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 7.04i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.54 - 0.485i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-2.37 - 1.37i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.76 - 10.3i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.61 + 4.30i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.20 - 13.7i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.692 - 1.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.62 - 2.15i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.26698814182020351365490021559, −8.985299596927166966474640498445, −8.296186775584278510267107542680, −7.33020122924065124056661467172, −6.52716528663506261651215696050, −5.57287221383807472180254629830, −4.44747765815088365527153432686, −3.66092499103507533816370538075, −2.72715027722276544066397158732, −0.59829530825396135880922579562,
1.86208515016338673438815842800, 3.17199116720766883732654241395, 4.18809792089206494276611420621, 4.81885119746636602700017015912, 6.15929317583168740329576927742, 7.05403637648985207916106971720, 7.53766405800366560562929840680, 8.891431795811547371573468285088, 9.584582877051113846451880999676, 10.62809640801017407509766240097
