L(s) = 1 | + (2.98 + 1.72i)2-s − 1.73i·3-s + (3.95 + 6.85i)4-s + 6.27i·5-s + (2.98 − 5.17i)6-s + (−1.92 + 1.10i)7-s + 13.5i·8-s − 2.99·9-s + (−10.8 + 18.7i)10-s + (0.342 − 0.197i)11-s + (11.8 − 6.85i)12-s + (7.58 + 4.37i)13-s − 7.66·14-s + 10.8·15-s + (−7.52 + 13.0i)16-s + (13.4 − 23.2i)17-s + ⋯ |
L(s) = 1 | + (1.49 + 0.863i)2-s − 0.577i·3-s + (0.989 + 1.71i)4-s + 1.25i·5-s + (0.498 − 0.863i)6-s + (−0.274 + 0.158i)7-s + 1.69i·8-s − 0.333·9-s + (−1.08 + 1.87i)10-s + (0.0311 − 0.0179i)11-s + (0.989 − 0.571i)12-s + (0.583 + 0.336i)13-s − 0.547·14-s + 0.724·15-s + (−0.470 + 0.814i)16-s + (0.789 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.40369 + 2.23310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40369 + 2.23310i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 67 | \( 1 + (48.2 - 46.4i)T \) |
good | 2 | \( 1 + (-2.98 - 1.72i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 - 6.27iT - 25T^{2} \) |
| 7 | \( 1 + (1.92 - 1.10i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.342 + 0.197i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.58 - 4.37i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-13.4 + 23.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-5.15 + 8.92i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.88i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (11.9 + 20.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (46.3 - 26.7i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-20.1 + 34.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-20.0 + 11.5i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 - 26.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (10.2 + 17.7i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 77.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-24.5 - 14.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 71 | \( 1 + (-22.0 - 38.2i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-26.8 + 46.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-108. + 62.4i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (49.1 - 85.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 5.71T + 7.92e3T^{2} \) |
| 97 | \( 1 + (23.0 + 13.2i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.74360231613542495469050096933, −11.75032675181502542581519119510, −10.99754549068089247329982826586, −9.393930239244742386256214465717, −7.70703835362463637559734191737, −7.04844689870014534719119866892, −6.26290432595597536943042461624, −5.26685636563976103330872038568, −3.64684154132541083225452029948, −2.69834990813542834825057269833,
1.46756598811568121996575283252, 3.39227963642098977399506330720, 4.21740644208003269209550115963, 5.32012434002905746405064209476, 6.04834457035996532722593009127, 8.079626116527604091236786131575, 9.305460850721747815914882006572, 10.37034268986062554681544490105, 11.18569304027908120770503234623, 12.35064850912204119639541262690