| 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
−10.92366328995786220242428290305, −10.62747816779255620494256984639, −9.274722699099827287897528234201, −7.43151995438147181631858316904, −6.58818428186730537829713182095, −6.02841196652625654807695072223, −4.68163660977248337926216022580, −3.43550175107326984696996157847, −2.68381192233424925461583322104, −1.15826622062639609016483216667,
3.08438239604520615565880729923, 4.02724465813380332245993713024, 4.98956438838434129671694685532, 5.71958758996511532242200770467, 6.23751065847974879391747401052, 7.83955762395822564480767152994, 8.637555077026218559077591797567, 9.379944426396562683661101154688, 10.96682946981153397466631076296, 11.90061902348158105007895249041
