L(s) = 1 | − 0.389i·2-s + 5.19·3-s + 15.8·4-s − 17.9·5-s − 2.02i·6-s − 47.2·7-s − 12.3i·8-s + 27·9-s + 7.00i·10-s + 197. i·11-s + 82.3·12-s + 176. i·13-s + 18.3i·14-s − 93.5·15-s + 248.·16-s + 486.·17-s + ⋯ |
L(s) = 1 | − 0.0972i·2-s + 0.577·3-s + 0.990·4-s − 0.719·5-s − 0.0561i·6-s − 0.964·7-s − 0.193i·8-s + 0.333·9-s + 0.0700i·10-s + 1.63i·11-s + 0.571·12-s + 1.04i·13-s + 0.0938i·14-s − 0.415·15-s + 0.971·16-s + 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.167934737\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167934737\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (1.64e3 - 3.07e3i)T \) |
good | 2 | \( 1 + 0.389iT - 16T^{2} \) |
| 5 | \( 1 + 17.9T + 625T^{2} \) |
| 7 | \( 1 + 47.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 197. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 176. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 486.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 56.2T + 1.30e5T^{2} \) |
| 23 | \( 1 - 848. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 275.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 843. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.85e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.47e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.75e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.43e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.11e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 115. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 49.5iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.08e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 209. iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.80e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.11e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 2.94e3iT - 8.85e7T^{2} \) |
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
−12.17263001316793562400229202053, −11.39897932849664111665764985714, −9.924207143427792161875697349826, −9.494559140510249733661831327206, −7.59509624981363470874641079847, −7.36689057572461014525262094988, −5.99934722557917350171577718208, −4.14274164139846845733046831559, −3.06277466036015598654451706631, −1.65626613254303885809169546706,
0.75327542977425079848788361952, 2.96051640509040109445353618480, 3.45589394141272657385435183092, 5.61981176122431066763337043098, 6.62835585907513802824423555247, 7.84632096173540158048070110026, 8.455985288588421511408907170737, 10.02374598477805968947790690887, 10.76504634026494972175671691077, 11.96227368449317147282186119122