L(s) = 1 | + (2.10 + 1.67i)2-s + (−0.574 + 1.63i)3-s + (1.16 + 5.09i)4-s + (−0.190 − 2.53i)5-s + (−3.94 + 2.47i)6-s + (−1.20 + 2.35i)7-s + (−3.76 + 7.80i)8-s + (−2.34 − 1.87i)9-s + (3.85 − 5.65i)10-s + (−1.07 + 0.420i)11-s + (−8.99 − 1.02i)12-s + (5.20 − 2.04i)13-s + (−6.47 + 2.93i)14-s + (4.25 + 1.14i)15-s + (−11.5 + 5.57i)16-s + (−2.91 + 0.899i)17-s + ⋯ |
L(s) = 1 | + (1.48 + 1.18i)2-s + (−0.331 + 0.943i)3-s + (0.581 + 2.54i)4-s + (−0.0850 − 1.13i)5-s + (−1.61 + 1.00i)6-s + (−0.454 + 0.890i)7-s + (−1.32 + 2.76i)8-s + (−0.780 − 0.625i)9-s + (1.21 − 1.78i)10-s + (−0.323 + 0.126i)11-s + (−2.59 − 0.295i)12-s + (1.44 − 0.566i)13-s + (−1.73 + 0.784i)14-s + (1.09 + 0.296i)15-s + (−2.89 + 1.39i)16-s + (−0.707 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.381119 + 2.50517i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.381119 + 2.50517i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.574 - 1.63i)T \) |
| 7 | \( 1 + (1.20 - 2.35i)T \) |
good | 2 | \( 1 + (-2.10 - 1.67i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (0.190 + 2.53i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.07 - 0.420i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-5.20 + 2.04i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (2.91 - 0.899i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-5.90 - 3.41i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.156 + 0.168i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.38 - 4.49i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 8.38iT - 31T^{2} \) |
| 37 | \( 1 + (-0.586 - 0.544i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-4.95 + 3.37i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-5.71 - 3.89i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (-1.77 + 2.22i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.44 + 6.94i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (8.75 - 4.21i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (10.4 + 2.38i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 + (6.75 - 1.54i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.40 + 1.72i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + (-0.328 + 0.837i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (0.304 - 0.0459i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (0.386 - 0.223i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.90061902348158105007895249041, −10.96682946981153397466631076296, −9.379944426396562683661101154688, −8.637555077026218559077591797567, −7.83955762395822564480767152994, −6.23751065847974879391747401052, −5.71958758996511532242200770467, −4.98956438838434129671694685532, −4.02724465813380332245993713024, −3.08438239604520615565880729923,
1.15826622062639609016483216667, 2.68381192233424925461583322104, 3.43550175107326984696996157847, 4.68163660977248337926216022580, 6.02841196652625654807695072223, 6.58818428186730537829713182095, 7.43151995438147181631858316904, 9.274722699099827287897528234201, 10.62747816779255620494256984639, 10.92366328995786220242428290305