L(s) = 1 | + (1.96 + 2.26i)3-s + (2.89 − 2.76i)4-s + (4.57 + 3.59i)7-s + (−1.28 + 8.90i)9-s + (11.9 + 1.14i)12-s + (5.06 − 7.10i)13-s + (0.761 − 15.9i)16-s + (−3.07 + 2.42i)19-s + (0.831 + 17.4i)21-s + (10.3 + 22.7i)25-s + (−22.7 + 14.5i)27-s + (23.1 − 2.21i)28-s + (12.6 + 17.7i)31-s + (20.8 + 29.3i)36-s + (−36.9 − 63.9i)37-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)3-s + (0.723 − 0.690i)4-s + (0.653 + 0.514i)7-s + (−0.142 + 0.989i)9-s + (0.995 + 0.0950i)12-s + (0.389 − 0.546i)13-s + (0.0475 − 0.998i)16-s + (−0.162 + 0.127i)19-s + (0.0395 + 0.830i)21-s + (0.415 + 0.909i)25-s + (−0.841 + 0.540i)27-s + (0.828 − 0.0790i)28-s + (0.408 + 0.573i)31-s + (0.580 + 0.814i)36-s + (−0.997 − 1.72i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.22093 + 0.500305i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22093 + 0.500305i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.96 - 2.26i)T \) |
| 67 | \( 1 + (58.0 + 33.3i)T \) |
good | 2 | \( 1 + (-2.89 + 2.76i)T^{2} \) |
| 5 | \( 1 + (-10.3 - 22.7i)T^{2} \) |
| 7 | \( 1 + (-4.57 - 3.59i)T + (11.5 + 47.6i)T^{2} \) |
| 11 | \( 1 + (-70.1 + 98.5i)T^{2} \) |
| 13 | \( 1 + (-5.06 + 7.10i)T + (-55.2 - 159. i)T^{2} \) |
| 17 | \( 1 + (-13.7 - 288. i)T^{2} \) |
| 19 | \( 1 + (3.07 - 2.42i)T + (85.1 - 350. i)T^{2} \) |
| 23 | \( 1 + (415. + 327. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-12.6 - 17.7i)T + (-314. + 908. i)T^{2} \) |
| 37 | \( 1 + (36.9 + 63.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (1.49e3 + 770. i)T^{2} \) |
| 43 | \( 1 + (28.5 - 8.39i)T + (1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + (-2.05e3 + 821. i)T^{2} \) |
| 53 | \( 1 + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (48.2 + 24.8i)T + (2.15e3 + 3.03e3i)T^{2} \) |
| 71 | \( 1 + (-239. + 5.03e3i)T^{2} \) |
| 73 | \( 1 + (121. + 62.4i)T + (3.09e3 + 4.34e3i)T^{2} \) |
| 79 | \( 1 + (-25.8 - 2.46i)T + (6.12e3 + 1.18e3i)T^{2} \) |
| 83 | \( 1 + (6.85e3 + 654. i)T^{2} \) |
| 89 | \( 1 + (1.12e3 + 7.84e3i)T^{2} \) |
| 97 | \( 1 + (-91.9 - 159. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.11000825847359057508598691841, −11.01787090392301580555717011253, −10.47532822787702262635315625638, −9.345156645686107224391346135634, −8.415151368995098618181711688012, −7.29448489724166711928664845759, −5.79332839378630361577182256010, −4.89282775406505566935206910103, −3.23418829690397668806159388138, −1.85651616025767959437945609766,
1.59342509524605268294879027383, 2.95882189096967002047157503327, 4.29451523518121364634986884549, 6.30665431954936751583005854459, 7.13356839909183538511605745069, 8.068424733300472781955048436026, 8.771111659374341917519886654449, 10.29953872718558293801977020534, 11.46815126135976722385140448123, 12.09065991490794943278375952368