L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.16 − 3.74i)5-s + (−2.22 − 1.42i)7-s − 0.999·8-s + (−2.16 − 3.74i)10-s + (1.88 + 3.26i)11-s − 13-s + (−2.35 + 1.21i)14-s + (−0.5 + 0.866i)16-s + (−1.44 − 2.50i)17-s + (0.0658 − 0.114i)19-s − 4.32·20-s + 3.76·22-s + (3.03 − 5.26i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.966 − 1.67i)5-s + (−0.841 − 0.539i)7-s − 0.353·8-s + (−0.683 − 1.18i)10-s + (0.567 + 0.983i)11-s − 0.277·13-s + (−0.628 + 0.324i)14-s + (−0.125 + 0.216i)16-s + (−0.351 − 0.608i)17-s + (0.0151 − 0.0261i)19-s − 0.966·20-s + 0.802·22-s + (0.633 − 1.09i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783814598\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783814598\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.22 + 1.42i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-2.16 + 3.74i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.88 - 3.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.44 + 2.50i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0658 + 0.114i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.03 + 5.26i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.377T + 29T^{2} \) |
| 31 | \( 1 + (1.77 + 3.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 6.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 + (5.84 - 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.36 + 2.36i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.68 - 4.64i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.22 - 10.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.963 - 1.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.77T + 71T^{2} \) |
| 73 | \( 1 + (-3.80 - 6.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0247 + 0.0428i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.30T + 83T^{2} \) |
| 89 | \( 1 + (0.416 - 0.720i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−9.306474960763427477760717669934, −8.548721568809577751213840667139, −7.28328410261666138009503329084, −6.38627495506291221364103835543, −5.54553094185923943963895285502, −4.60352417494716988036433577419, −4.21817741089653385647181496329, −2.71305739240679573032351964743, −1.68398681636792259394104119551, −0.58668454537198843405594404601,
1.99698011281199031452093809935, 3.22679469457643999938677697834, 3.45696003355228676510578371129, 5.18920697626899951448531091095, 5.96870456146611555389980285671, 6.53446714206538253102171426451, 6.96173248079589206901794683876, 8.078841447622844501467962158383, 9.103784666533564758228144747147, 9.666186594136393708784081547131