| L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.826 − 0.300i)5-s + (−1.09 − 1.89i)7-s + (0.173 + 0.984i)9-s + (−0.0812 + 0.140i)11-s + (0.581 − 0.487i)13-s + (−0.826 − 0.300i)15-s + (0.539 − 3.05i)17-s + (−2.77 − 3.35i)19-s + (−0.379 + 2.15i)21-s + (1.21 + 0.441i)23-s + (−3.23 + 2.71i)25-s + (0.500 − 0.866i)27-s + (−1.13 − 6.41i)29-s + (0.479 + 0.829i)31-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.371i)3-s + (0.369 − 0.134i)5-s + (−0.412 − 0.715i)7-s + (0.0578 + 0.328i)9-s + (−0.0244 + 0.0424i)11-s + (0.161 − 0.135i)13-s + (−0.213 − 0.0776i)15-s + (0.130 − 0.741i)17-s + (−0.637 − 0.770i)19-s + (−0.0827 + 0.469i)21-s + (0.252 + 0.0920i)23-s + (−0.647 + 0.543i)25-s + (0.0962 − 0.166i)27-s + (−0.210 − 1.19i)29-s + (0.0860 + 0.149i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.380806 - 0.841970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380806 - 0.841970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (2.77 + 3.35i)T \) |
| good | 5 | \( 1 + (-0.826 + 0.300i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0812 - 0.140i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.581 + 0.487i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.539 + 3.05i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 0.441i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.13 + 6.41i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.479 - 0.829i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + (8.11 + 6.81i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.166 - 0.0605i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.602 - 3.41i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (7.83 + 2.84i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.482 + 2.73i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.79 + 2.47i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.184 + 1.04i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.77 + 1.73i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.72 - 1.44i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.01 - 3.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.55 + 14.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.9 + 10.0i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.63 - 9.27i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.835691954225003538141970577895, −9.072535565189332411995962647181, −7.962800367920204892612832014311, −7.12606093827210158249149632005, −6.42229899928949838657070188334, −5.46715402335344388622429710712, −4.52456741605647699539850553082, −3.33291841155477083199561106358, −1.96275492818626141448336595777, −0.45127061799494670711253750165,
1.73579350098789440633448833991, 3.08223815905244983724436294723, 4.14915968120378601622293514389, 5.29755353569939446277988069589, 6.07353418931826029497743160913, 6.68899501514532183507347488276, 8.018627475202872402148117456348, 8.820668859858642979862119661850, 9.661225502356515891832788680432, 10.37249278918257437919959794781
