| 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 | \( 1 + (-8.35e3 - 8.35e3i)T + 8.58e9iT^{2} \) |
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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
−10.55454322611987950644143368650, −9.581025761560521790239713478677, −8.905022633730198339316473657554, −7.18642627304327584090020373553, −6.81547727067850589181635982439, −5.56160166208490435097191610695, −4.29813809261597142387430400347, −3.59326297444736901753476802862, −2.40342139258418503307483616178, −0.917134710424190480882321025706,
0.24263848711831313397850728984, 2.46304031857370567161222749761, 3.09106280475709562499250396001, 4.60855868839256534833575563736, 5.49395002602036288948630734672, 6.43006325233781689908607129738, 7.18680561060220485259071555817, 8.391438132375170378598055240209, 9.249259673817992365955387898953, 9.997613347609387304171276651366
