| L(s) = 1 | + (1.30 − 3.15i)2-s + (2.10 − 3.14i)3-s + (−5.40 − 5.40i)4-s + (−0.793 − 3.98i)5-s + (−7.17 − 10.7i)6-s + (−1.40 + 7.05i)7-s + (−11.4 + 4.74i)8-s + (−2.03 − 4.91i)9-s + (−13.6 − 2.70i)10-s + (−8.47 + 5.66i)11-s + (−28.3 + 5.63i)12-s + (6.16 − 6.16i)13-s + (20.4 + 13.6i)14-s + (−14.2 − 5.88i)15-s + 11.7i·16-s + ⋯ |
| L(s) = 1 | + (0.652 − 1.57i)2-s + (0.700 − 1.04i)3-s + (−1.35 − 1.35i)4-s + (−0.158 − 0.797i)5-s + (−1.19 − 1.78i)6-s + (−0.200 + 1.00i)7-s + (−1.43 + 0.593i)8-s + (−0.226 − 0.546i)9-s + (−1.36 − 0.270i)10-s + (−0.770 + 0.515i)11-s + (−2.36 + 0.469i)12-s + (0.474 − 0.474i)13-s + (1.45 + 0.973i)14-s + (−0.947 − 0.392i)15-s + 0.736i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.755668 + 2.24442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.755668 + 2.24442i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-1.30 + 3.15i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-2.10 + 3.14i)T + (-3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (0.793 + 3.98i)T + (-23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (1.40 - 7.05i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (8.47 - 5.66i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (-6.16 + 6.16i)T - 169iT^{2} \) |
| 19 | \( 1 + (-12.4 + 30.1i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (0.682 + 1.02i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-22.3 + 4.44i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (22.9 + 15.3i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (23.6 - 35.4i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-4.21 + 21.1i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-9.49 - 22.9i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-44.1 + 44.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (32.5 - 78.5i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-36.1 + 14.9i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-7.26 - 1.44i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 99.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (2.61 - 3.91i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-20.0 - 100. i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-48.9 + 32.6i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (83.6 + 34.6i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-5.23 - 5.23i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-95.4 + 18.9i)T + (8.69e3 - 3.60e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.36557297785878018419316217609, −10.24852468182062317106898247329, −9.112893989129090999380766280398, −8.471893411227281180969566316825, −7.21246763978090816453043390186, −5.53972543530431269309780708200, −4.63279025822790881518701755673, −2.98301303368836554553820647861, −2.29582540638403077623480461760, −0.933179822208524103014089174444,
3.37832228545282436406427353198, 3.87685083585292005831780603319, 5.09887049033606297970728808229, 6.28618217585740567651641987875, 7.26242181910861890224327058355, 8.050821578916714618306072800706, 9.037146525659326061552258934043, 10.23232609791926053509442319085, 10.86658631497818427494576591991, 12.53081372428999847872273095346
