| L(s) = 1 | + (1.34 + 1.09i)3-s + 3.88·5-s + (−3.35 + 1.93i)7-s + (0.618 + 2.93i)9-s + (−0.627 − 1.08i)11-s + (0.212 + 0.122i)13-s + (5.22 + 4.23i)15-s + (2.38 + 1.37i)17-s + (−2.25 + 3.90i)19-s + (−6.63 − 1.05i)21-s + (2.10 + 1.21i)23-s + 10.0·25-s + (−2.37 + 4.62i)27-s + (−3.32 + 1.91i)29-s + (4.36 − 2.51i)31-s + ⋯ |
| L(s) = 1 | + (0.776 + 0.629i)3-s + 1.73·5-s + (−1.26 + 0.733i)7-s + (0.206 + 0.978i)9-s + (−0.189 − 0.327i)11-s + (0.0589 + 0.0340i)13-s + (1.34 + 1.09i)15-s + (0.577 + 0.333i)17-s + (−0.517 + 0.895i)19-s + (−1.44 − 0.230i)21-s + (0.439 + 0.253i)23-s + 2.01·25-s + (−0.456 + 0.889i)27-s + (−0.617 + 0.356i)29-s + (0.783 − 0.452i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93806 + 1.29816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93806 + 1.29816i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.34 - 1.09i)T \) |
| 67 | \( 1 + (-0.688 - 8.15i)T \) |
| good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + (3.35 - 1.93i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.627 + 1.08i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.212 - 0.122i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.38 - 1.37i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.25 - 3.90i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.10 - 1.21i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.32 - 1.91i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.36 + 2.51i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.49 + 7.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.59 + 6.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 4.98iT - 43T^{2} \) |
| 47 | \( 1 + (4.99 - 2.88i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 13.5iT - 59T^{2} \) |
| 61 | \( 1 + (6.26 + 3.61i)T + (30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (4.57 - 2.64i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.33 - 2.31i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 5.96i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.48 - 2.59i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.5iT - 89T^{2} \) |
| 97 | \( 1 + (7.87 + 4.54i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.16472548176064151176978258705, −9.506310014138734926171069007531, −9.081487318936528154107999706761, −8.140236493506951865247432850328, −6.77612949324823962237702338654, −5.83537193518848590465473313732, −5.37267460334757843153318936755, −3.77770241461858738069913287389, −2.80397241698064756452505102841, −1.94219515562104036578376180147,
1.13416567410049931786948112232, 2.48200420856773061937249881680, 3.19548939221862222203533755742, 4.70328510665114374786051024705, 6.06163289188558327381457415388, 6.58796179735824296270957361416, 7.32615823561302538141265830093, 8.556935027335578400539436350651, 9.460158586722961720082232459430, 9.850642414175082891121340029354
