L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (0.242 + 0.140i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.663 − 0.383i)11-s + 0.999·12-s + (−2.77 + 2.30i)13-s − 0.280·14-s + (−0.5 − 0.866i)16-s + (−1.10 + 1.92i)17-s − 0.999i·18-s + (−4.57 − 2.63i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (0.0917 + 0.0529i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.200 − 0.115i)11-s + 0.288·12-s + (−0.769 + 0.639i)13-s − 0.0749·14-s + (−0.125 − 0.216i)16-s + (−0.269 + 0.465i)17-s − 0.235i·18-s + (−1.04 − 0.605i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1516531863\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1516531863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.77 - 2.30i)T \) |
good | 7 | \( 1 + (-0.242 - 0.140i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.663 + 0.383i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.10 - 1.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.57 + 2.63i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.725 + 1.25i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.03 + 5.24i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.28iT - 31T^{2} \) |
| 37 | \( 1 + (3.07 - 1.77i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.92 + 1.10i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.80 - 3.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.06iT - 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 + (4.32 + 2.49i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.773 - 1.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.43 - 3.71i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.09 + 1.21i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 4.42iT - 83T^{2} \) |
| 89 | \( 1 + (4.60 - 2.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.09 - 0.633i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.619352450979025719542725154246, −8.765833967985624455987138536424, −8.366152774443347225142336895309, −7.33929458288012774180379814132, −6.64829198920316787324658853485, −5.78525393822921264553083869960, −4.74331190916128823422594001665, −4.06050773972134710350092765257, −2.72744006136172788069071055090, −1.77630710467572904469153441925,
0.06078804906728435947556577575, 1.54118430960403340960725005630, 2.46227542496620694498954978439, 3.42801271512830144911811162478, 4.48369826594362356740238711263, 5.60473694554956697947349865760, 6.55452290418272474137700395738, 7.39017493084279518533342238613, 7.895544703351618691118598787064, 8.786131378514742654952125974363