| L(s) = 1 | + (2.44 − 5.90i)2-s + (0.660 − 0.988i)3-s + (−17.5 − 17.5i)4-s + (5.91 + 29.7i)5-s + (−4.22 − 6.31i)6-s + (9.51 − 47.8i)7-s + (−52.3 + 21.6i)8-s + (30.4 + 73.5i)9-s + (189. + 37.7i)10-s + (9.22 − 6.16i)11-s + (−29.0 + 5.76i)12-s + (−175. + 175. i)13-s + (−259. − 173. i)14-s + (33.2 + 13.7i)15-s − 35.4i·16-s + (−53.0 − 284. i)17-s + ⋯ |
| L(s) = 1 | + (0.611 − 1.47i)2-s + (0.0733 − 0.109i)3-s + (−1.09 − 1.09i)4-s + (0.236 + 1.18i)5-s + (−0.117 − 0.175i)6-s + (0.194 − 0.976i)7-s + (−0.818 + 0.339i)8-s + (0.376 + 0.907i)9-s + (1.89 + 0.377i)10-s + (0.0762 − 0.0509i)11-s + (−0.201 + 0.0400i)12-s + (−1.03 + 1.03i)13-s + (−1.32 − 0.883i)14-s + (0.147 + 0.0612i)15-s − 0.138i·16-s + (−0.183 − 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0434 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0434 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.10299 - 1.15201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10299 - 1.15201i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (53.0 + 284. i)T \) |
| good | 2 | \( 1 + (-2.44 + 5.90i)T + (-11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (-0.660 + 0.988i)T + (-30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (-5.91 - 29.7i)T + (-577. + 239. i)T^{2} \) |
| 7 | \( 1 + (-9.51 + 47.8i)T + (-2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (-9.22 + 6.16i)T + (5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (175. - 175. i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (174. - 422. i)T + (-9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (402. + 602. i)T + (-1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (259. - 51.5i)T + (6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (-1.17e3 - 784. i)T + (3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-753. + 1.12e3i)T + (-7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (-143. + 719. i)T + (-2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (415. + 1.00e3i)T + (-2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-1.96e3 + 1.96e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (747. - 1.80e3i)T + (-5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (837. - 346. i)T + (8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (3.97e3 + 790. i)T + (1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 - 4.81e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-596. + 893. i)T + (-9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-1.34e3 - 6.74e3i)T + (-2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (-5.44e3 + 3.63e3i)T + (1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (-218. - 90.5i)T + (3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-354. - 354. i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (5.29e3 - 1.05e3i)T + (8.17e7 - 3.38e7i)T^{2} \) |
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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
−18.46662904447570973541650604814, −16.64453441218623039023367941374, −14.27446322859373457557670177998, −13.86734187220751563022462238014, −12.18467496692771486515077246335, −10.78559664546371893127347719739, −10.08102868069896650955529139451, −7.17708803183157519825247177139, −4.35757023817325017715569384691, −2.31387491581891586246523953732,
4.72925067457402506342289846047, 6.05573359530015924901385697558, 8.050938923715366503995931947762, 9.345757866696385132455260705042, 12.30332717074791968563583304659, 13.23778708088221938349795939835, 15.07222651400272758656461526842, 15.46709836855644803903114615399, 17.03986195128560684808451878601, 17.76057280097477385809554150460
