| L(s) = 1 | − 256·4-s + (−144.5 − 608. i)5-s − 4.77e3i·7-s − 6.56e3·9-s − 2.50e4·11-s + 6.55e4·16-s − 1.61e5i·17-s + 1.30e5·19-s + (3.69e4 + 1.55e5i)20-s − 1.65e5i·23-s + (−3.48e5 + 1.75e5i)25-s + 1.22e6i·28-s + (−2.90e6 + 6.89e5i)35-s + 1.67e6·36-s + 3.92e6i·43-s + 6.40e6·44-s + ⋯ |
| L(s) = 1 | − 4-s + (−0.231 − 0.972i)5-s − 1.98i·7-s − 9-s − 1.70·11-s + 16-s − 1.93i·17-s + 19-s + (0.231 + 0.972i)20-s − 0.590i·23-s + (−0.893 + 0.449i)25-s + 1.98i·28-s + (−1.93 + 0.459i)35-s + 36-s + 1.14i·43-s + 1.70·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.321777 + 0.407205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.321777 + 0.407205i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (144.5 + 608. i)T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 + 256T^{2} \) |
| 3 | \( 1 + 6.56e3T^{2} \) |
| 7 | \( 1 + 4.77e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.50e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.61e5iT - 6.97e9T^{2} \) |
| 23 | \( 1 + 1.65e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.51e12T^{2} \) |
| 41 | \( 1 - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.92e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 5.12e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.76e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.60e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.51e15T^{2} \) |
| 83 | \( 1 + 7.12e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.83e15T^{2} \) |
| 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
−11.61349287628487016043739062860, −10.38395981365101916998289684155, −9.441254722039633835302120698325, −8.157847498349727484338907977679, −7.44001629790758694233100463507, −5.27362088235628917902224698127, −4.62878272543920770581143340095, −3.20982920975004536173935117487, −0.69765245666182102899342331833, −0.24747955139998980848507544441,
2.35716366250492988927084851065, 3.35738187366264282516552921442, 5.35254455010733943196390208267, 5.88088172116648961382857703210, 7.966856900047827985253232132310, 8.598847084065808033360108709647, 9.841641951633655390877932139501, 11.00049870000897647136142706365, 12.14200549126359614842940250795, 13.06482801214118280544276080641
