| L(s) = 1 | + (15.7 − 2.54i)2-s + (21.5 + 7.01i)3-s + (243. − 80.2i)4-s + (−500. + 374. i)5-s + (358. + 55.9i)6-s − 2.75e3i·7-s + (3.63e3 − 1.88e3i)8-s + (−4.89e3 − 3.55e3i)9-s + (−6.95e3 + 7.18e3i)10-s + (7.97e3 + 1.09e4i)11-s + (5.80e3 − 28.4i)12-s + (−3.24e4 − 2.35e4i)13-s + (−7.00e3 − 4.35e4i)14-s + (−1.34e4 + 4.56e3i)15-s + (5.26e4 − 3.90e4i)16-s + (3.58e4 + 1.10e5i)17-s + ⋯ |
| L(s) = 1 | + (0.987 − 0.158i)2-s + (0.266 + 0.0865i)3-s + (0.949 − 0.313i)4-s + (−0.801 + 0.598i)5-s + (0.276 + 0.0431i)6-s − 1.14i·7-s + (0.887 − 0.460i)8-s + (−0.745 − 0.541i)9-s + (−0.695 + 0.718i)10-s + (0.544 + 0.749i)11-s + (0.280 − 0.00137i)12-s + (−1.13 − 0.824i)13-s + (−0.182 − 1.13i)14-s + (−0.265 + 0.0900i)15-s + (0.803 − 0.595i)16-s + (0.428 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.335333 - 1.39614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335333 - 1.39614i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.7 + 2.54i)T \) |
| 5 | \( 1 + (500. - 374. i)T \) |
| good | 3 | \( 1 + (-21.5 - 7.01i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 2.75e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-7.97e3 - 1.09e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (3.24e4 + 2.35e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-3.58e4 - 1.10e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (2.17e5 - 7.06e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (2.18e5 + 3.00e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-2.25e5 + 6.94e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (7.34e5 - 2.38e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (2.85e6 + 2.07e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.24e6 - 1.62e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 3.06e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.42e5 + 4.63e4i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.40e5 + 4.31e5i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (5.82e6 - 8.02e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (5.10e6 - 3.70e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.94e7 + 9.56e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (2.76e6 + 8.99e5i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (8.00e6 - 5.81e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.10e7 + 6.84e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (7.43e6 - 2.41e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-1.96e7 + 1.42e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (2.17e7 - 6.68e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
| 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
−12.14833609265549099660274392805, −10.70912868512626395978750426688, −10.21766433211103524998162215757, −8.168309164496879131543245576603, −7.15321358590557113748089992112, −6.11069626428909290587135462277, −4.27688741576330239053261834011, −3.67696505546101193704260813459, −2.23018832937848570040082063339, −0.24778299884981642768358387112,
2.03629921264317147676808354259, 3.20789945643290452062896641732, 4.68347836096723602342694706250, 5.59222650179835765087650492347, 7.04985833722267762995065102220, 8.268205512665087014657631739003, 9.154111975220138303443245258863, 11.20981038799188841989711823979, 11.83847619914919977529428709073, 12.59439354867700160298534718643
