L(s) = 1 | + (−1.31 + 0.520i)2-s + (−1.07 + 1.85i)3-s + (1.45 − 1.36i)4-s + (−2.26 + 1.30i)5-s + (0.443 − 2.99i)6-s + (−1.20 + 2.55i)8-s + (−0.792 − 1.37i)9-s + (2.29 − 2.89i)10-s + (−3.42 − 1.97i)11-s + (0.974 + 4.17i)12-s − 1.08i·13-s − 5.59i·15-s + (0.257 − 3.99i)16-s + (0.274 + 0.158i)17-s + (1.75 + 1.39i)18-s + (−2.58 − 4.47i)19-s + ⋯ |
L(s) = 1 | + (−0.929 + 0.367i)2-s + (−0.618 + 1.07i)3-s + (0.729 − 0.684i)4-s + (−1.01 + 0.584i)5-s + (0.181 − 1.22i)6-s + (−0.426 + 0.904i)8-s + (−0.264 − 0.457i)9-s + (0.726 − 0.915i)10-s + (−1.03 − 0.596i)11-s + (0.281 + 1.20i)12-s − 0.300i·13-s − 1.44i·15-s + (0.0642 − 0.997i)16-s + (0.0665 + 0.0384i)17-s + (0.414 + 0.328i)18-s + (−0.593 − 1.02i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0563541 - 0.124671i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0563541 - 0.124671i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.31 - 0.520i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.07 - 1.85i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.26 - 1.30i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.42 + 1.97i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.08iT - 13T^{2} \) |
| 17 | \( 1 + (-0.274 - 0.158i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.58 + 4.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 1.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + (3.02 - 5.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.23iT - 43T^{2} \) |
| 47 | \( 1 + (-2.14 - 3.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.24 - 9.08i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.61 + 9.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.68 - 2.70i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.83 - 1.63i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (12.1 + 7.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.85 - 3.95i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + (5.18 - 2.99i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.23iT - 97T^{2} \) |
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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
−12.98760234864736590822312040988, −11.42239571362000974204040050989, −11.00507125558405521941046977224, −10.36908716794714181899233052377, −9.229948036720923463791768525554, −8.085848468443012741777134592960, −7.19607834364389673045414259079, −5.84148380977958161177557905034, −4.71724587206422995643877358712, −3.06632332040407790522481403450,
0.16540971228838015836457383034, 1.93270132770834664519279135516, 3.95249541212622586019998598014, 5.75360311655284005132553882594, 7.20352011306458820567884514324, 7.67540422029561932133267118482, 8.680559945976215339521345544622, 9.965753958410306125744660239034, 11.13137018450273334613577138561, 11.85743890465899265813814595502