L(s) = 1 | + (1.23 + 2.14i)2-s + (−2.05 + 3.56i)4-s + 2.75·5-s + (2.34 − 4.06i)7-s − 5.22·8-s + (3.40 + 5.89i)10-s + (−2.47 + 4.28i)11-s + (0.0567 + 0.0983i)13-s + 11.6·14-s + (−2.34 − 4.06i)16-s + (2.61 + 4.52i)17-s + (−0.209 − 0.363i)19-s + (−5.66 + 9.80i)20-s − 12.2·22-s + (−1.81 − 3.14i)23-s + ⋯ |
L(s) = 1 | + (0.874 + 1.51i)2-s + (−1.02 + 1.78i)4-s + 1.23·5-s + (0.887 − 1.53i)7-s − 1.84·8-s + (1.07 + 1.86i)10-s + (−0.745 + 1.29i)11-s + (0.0157 + 0.0272i)13-s + 3.10·14-s + (−0.586 − 1.01i)16-s + (0.633 + 1.09i)17-s + (−0.0481 − 0.0833i)19-s + (−1.26 + 2.19i)20-s − 2.60·22-s + (−0.378 − 0.655i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44058 + 2.35717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44058 + 2.35717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 67 | \( 1 + (1.25 - 8.08i)T \) |
good | 2 | \( 1 + (-1.23 - 2.14i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 + (-2.34 + 4.06i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.47 - 4.28i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0567 - 0.0983i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.61 - 4.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.209 + 0.363i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.81 + 3.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.19 + 5.52i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.63 + 9.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.42 + 7.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + (1.67 - 2.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.66T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + (0.347 + 0.601i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (1.81 - 3.14i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.36 + 9.29i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.17 + 12.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.140 + 0.243i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 + (-9.46 - 16.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.48742655195443515329552174148, −10.31341309458413378708881642900, −8.902806360383514410764604840364, −7.77892704760956988087576488113, −7.35651536876252088129875497776, −6.39547955481668395269230361540, −5.47407701684421727317394474193, −4.68141811380407086260304437190, −3.87446542797634161852652919483, −1.90407883718964159636396143037,
1.47311006090642491810496445509, 2.48980138563446301179975462967, 3.18542318145423039768786316541, 4.95902498594905646969267291728, 5.43148222013848405567970561410, 6.06173512349219540519209783546, 8.034985149281749096738846507248, 9.036326986257648281540915197735, 9.720509271054414492103929283767, 10.58374456739144225930120317596