L(s) = 1 | + (0.724 − 1.57i)3-s + (1 − 1.73i)4-s + 2.82i·5-s + (3.12 + 1.80i)7-s + (−1.94 − 2.28i)9-s + (3.11 + 1.14i)11-s + (−2 − 2.82i)12-s + (3.12 + 1.80i)13-s + (4.44 + 2.04i)15-s + (−1.99 − 3.46i)16-s + (−2.54 + 4.41i)17-s + (−6.24 − 3.60i)19-s + (4.89 + 2.82i)20-s + (5.09 − 3.60i)21-s + (1.22 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.418 − 0.908i)3-s + (0.5 − 0.866i)4-s + 1.26i·5-s + (1.18 + 0.681i)7-s + (−0.649 − 0.760i)9-s + (0.938 + 0.345i)11-s + (−0.577 − 0.816i)12-s + (0.866 + 0.499i)13-s + (1.14 + 0.529i)15-s + (−0.499 − 0.866i)16-s + (−0.618 + 1.07i)17-s + (−1.43 − 0.827i)19-s + (1.09 + 0.632i)20-s + (1.11 − 0.786i)21-s + (0.255 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.89494 - 0.500772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89494 - 0.500772i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.724 + 1.57i)T \) |
| 11 | \( 1 + (-3.11 - 1.14i)T \) |
| 13 | \( 1 + (-3.12 - 1.80i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + (-3.12 - 1.80i)T + (3.5 + 6.06i)T^{2} \) |
| 17 | \( 1 + (2.54 - 4.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.24 + 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.54 + 4.41i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 + 8.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.12 + 1.80i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + (-3.67 - 2.12i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.12 - 1.80i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.44 + 1.41i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 3.60iT - 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 + (2.44 - 1.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.24160902126431458748605396292, −10.48225650399751837107838906389, −9.048580201049640688720068926963, −8.421022970832591347032313558700, −7.03923729808204186478804526717, −6.58800664428096279015314492227, −5.75569678485003744959185347471, −4.06453753758798973073707821425, −2.35741776147607944689051126225, −1.74392190924411688422833733555,
1.64386923129970370788706601215, 3.44584298042806483049951223756, 4.32457579862894571796930517625, 5.09240981749246338586608450479, 6.62606414980759346745050208688, 8.058582484783778486758595831076, 8.431502387652884506857931243553, 9.125052690439528879647916363550, 10.48820247417600445477252086263, 11.26408863378996542174381685324