L(s) = 1 | + (2.82 + 2.82i)2-s + 16.0i·4-s + (−158. + 158. i)7-s + (−45.2 + 45.2i)8-s + 147. i·11-s + (−516. − 516. i)13-s − 897.·14-s − 256.·16-s + (−1.09e3 − 1.09e3i)17-s + 1.17e3i·19-s + (−415. + 415. i)22-s + (3.13e3 − 3.13e3i)23-s − 2.92e3i·26-s + (−2.53e3 − 2.53e3i)28-s + 1.86e3·29-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (−1.22 + 1.22i)7-s + (−0.250 + 0.250i)8-s + 0.366i·11-s + (−0.847 − 0.847i)13-s − 1.22·14-s − 0.250·16-s + (−0.917 − 0.917i)17-s + 0.747i·19-s + (−0.183 + 0.183i)22-s + (1.23 − 1.23i)23-s − 0.847i·26-s + (−0.611 − 0.611i)28-s + 0.411·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.032586024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.032586024\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 - 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (158. - 158. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 147. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (516. + 516. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.09e3 + 1.09e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.17e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-3.13e3 + 3.13e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 1.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.68e3 - 1.68e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 9.09e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.37e4 - 1.37e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-887. - 887. i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.63e4 + 1.63e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 957.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.57e4 + 3.57e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.59e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (5.28e4 + 5.28e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 1.63e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.95e4 + 6.95e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.33e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.35e3 + 8.35e3i)T - 8.58e9iT^{2} \) |
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
−9.997613347609387304171276651366, −9.249259673817992365955387898953, −8.391438132375170378598055240209, −7.18680561060220485259071555817, −6.43006325233781689908607129738, −5.49395002602036288948630734672, −4.60855868839256534833575563736, −3.09106280475709562499250396001, −2.46304031857370567161222749761, −0.24263848711831313397850728984,
0.917134710424190480882321025706, 2.40342139258418503307483616178, 3.59326297444736901753476802862, 4.29813809261597142387430400347, 5.56160166208490435097191610695, 6.81547727067850589181635982439, 7.18642627304327584090020373553, 8.905022633730198339316473657554, 9.581025761560521790239713478677, 10.55454322611987950644143368650