| L(s) = 1 | + (0.226 − 1.39i)2-s + (−0.295 + 0.894i)3-s + (−1.89 − 0.631i)4-s + (0.910 − 2.35i)5-s + (1.18 + 0.614i)6-s + (1.81 − 3.02i)7-s + (−1.31 + 2.50i)8-s + (1.69 + 1.25i)9-s + (−3.08 − 1.80i)10-s + (0.0911 + 0.738i)11-s + (1.12 − 1.51i)12-s + (0.0129 − 0.526i)13-s + (−3.81 − 3.21i)14-s + (1.84 + 1.51i)15-s + (3.20 + 2.39i)16-s + (−1.06 − 1.30i)17-s + ⋯ |
| L(s) = 1 | + (0.159 − 0.987i)2-s + (−0.170 + 0.516i)3-s + (−0.948 − 0.315i)4-s + (0.407 − 1.05i)5-s + (0.482 + 0.251i)6-s + (0.684 − 1.14i)7-s + (−0.463 + 0.886i)8-s + (0.565 + 0.419i)9-s + (−0.976 − 0.570i)10-s + (0.0274 + 0.222i)11-s + (0.324 − 0.436i)12-s + (0.00358 − 0.145i)13-s + (−1.01 − 0.858i)14-s + (0.475 + 0.390i)15-s + (0.800 + 0.599i)16-s + (−0.258 − 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.627827 - 1.30288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627827 - 1.30288i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.226 + 1.39i)T \) |
| good | 3 | \( 1 + (0.295 - 0.894i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (-0.910 + 2.35i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (-1.81 + 3.02i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.0911 - 0.738i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (-0.0129 + 0.526i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (1.06 + 1.30i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (6.10 + 4.29i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-2.24 + 4.74i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (-3.30 + 0.915i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (3.39 - 0.674i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.872 + 3.88i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (-2.86 + 2.59i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.0319 + 0.0634i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (-0.210 - 0.112i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (0.706 + 0.195i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (14.6 - 0.360i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (-10.8 - 9.32i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-12.8 + 0.946i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (1.08 - 7.28i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (-5.82 - 9.72i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (-6.61 - 2.00i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (3.11 - 13.8i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (-1.97 - 4.17i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 6.71i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.80884876324437937750340796014, −9.882250304027289976411661548786, −9.031167502436678061212477450086, −8.235400973233637503313803090822, −6.94696050335526898393819847998, −5.31081768115814123515776658252, −4.58899312616755277659408962413, −4.11485601245949700255055744591, −2.21671342912885817780112772031, −0.901473111861964050540125841220,
1.94058661133321735449608329418, 3.49563002378739447865660328243, 4.85390761462181897236851382050, 6.09361584094144202179622405073, 6.39742844761102224308914525687, 7.46633625204764839881628747004, 8.342988829296567737391698326036, 9.228437435550685045220609727665, 10.22178886820391394655701947875, 11.27796657463069871725899803142
