L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.227 − 0.393i)5-s + (−0.227 + 2.63i)7-s − 0.999·8-s + (−0.227 − 0.393i)10-s + (0.5 + 0.866i)11-s + 0.909·13-s + (2.16 + 1.51i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−1.94 + 3.36i)19-s − 0.454·20-s + 0.999·22-s + (4.16 − 7.22i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.101 − 0.176i)5-s + (−0.0859 + 0.996i)7-s − 0.353·8-s + (−0.0719 − 0.124i)10-s + (0.150 + 0.261i)11-s + 0.252·13-s + (0.579 + 0.404i)14-s + (−0.125 + 0.216i)16-s + (0.121 + 0.210i)17-s + (−0.445 + 0.771i)19-s − 0.101·20-s + 0.213·22-s + (0.869 − 1.50i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.938157522\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.938157522\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.227 - 2.63i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.227 + 0.393i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 0.909T + 13T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.94 - 3.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.16 + 7.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + (-4.79 - 8.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.94 - 6.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 + (-0.169 + 0.292i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.454 - 0.787i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.48 - 6.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.02 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.89 - 3.28i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 + (5.33 + 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.68 - 4.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 + (5.54 - 9.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−9.649044413000977834842627132867, −8.718753309095067960600363320794, −8.373098125662518173153687029966, −6.90100745126282702107634942755, −6.18538100385252059125607577411, −5.23727041209450013897514493113, −4.53453936305293463289359683384, −3.32835945909119646539991780145, −2.46337452148674481529389724608, −1.27280395697751210010120275782,
0.811291647395531261001553647214, 2.61022382274300642236050991269, 3.72369387986976542221537858866, 4.46059525284850912912553513194, 5.47726888508293089062944788536, 6.38015057413996319409498732364, 7.10488152462583994994607882377, 7.74112982497954375156088542897, 8.702878713133249219418507037858, 9.479101817037778419385868223014