| L(s) = 1 | + (0.495 + 3.12i)2-s + (1.54 + 0.786i)3-s + (−5.74 + 1.86i)4-s + (−4.93 + 0.775i)5-s + (−1.69 + 5.21i)6-s + (1.18 + 1.18i)7-s + (−2.92 − 5.74i)8-s + (1.76 + 2.42i)9-s + (−4.87 − 15.0i)10-s + (15.6 + 11.3i)11-s + (−10.3 − 1.63i)12-s + (3.58 − 22.6i)13-s + (−3.12 + 4.30i)14-s + (−8.23 − 2.68i)15-s + (−3.00 + 2.17i)16-s + (8.00 − 4.07i)17-s + ⋯ |
| L(s) = 1 | + (0.247 + 1.56i)2-s + (0.514 + 0.262i)3-s + (−1.43 + 0.466i)4-s + (−0.987 + 0.155i)5-s + (−0.282 + 0.869i)6-s + (0.169 + 0.169i)7-s + (−0.366 − 0.718i)8-s + (0.195 + 0.269i)9-s + (−0.487 − 1.50i)10-s + (1.42 + 1.03i)11-s + (−0.860 − 0.136i)12-s + (0.275 − 1.74i)13-s + (−0.223 + 0.307i)14-s + (−0.548 − 0.179i)15-s + (−0.187 + 0.136i)16-s + (0.470 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.421903 + 1.36509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421903 + 1.36509i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.54 - 0.786i)T \) |
| 5 | \( 1 + (4.93 - 0.775i)T \) |
| good | 2 | \( 1 + (-0.495 - 3.12i)T + (-3.80 + 1.23i)T^{2} \) |
| 7 | \( 1 + (-1.18 - 1.18i)T + 49iT^{2} \) |
| 11 | \( 1 + (-15.6 - 11.3i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-3.58 + 22.6i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-8.00 + 4.07i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-1.92 - 0.624i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (9.47 - 1.50i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-6.89 + 2.23i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-6.18 + 19.0i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (6.10 + 0.967i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (42.5 - 30.9i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (37.0 - 37.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-31.2 + 61.4i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (68.2 + 34.7i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-3.79 - 5.21i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-12.5 - 9.12i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-109. + 55.7i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-24.4 - 75.1i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (17.2 - 2.73i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-11.5 + 3.74i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (35.3 + 69.3i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (46.9 - 64.6i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (7.42 - 14.5i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−15.05090025666111505790119955165, −14.21433471742322178436508017543, −12.87538661619414735724363617833, −11.62978861752272476009950232379, −9.923796259128158001526195938038, −8.482268251539424112785499281833, −7.75682462419600802819953112814, −6.64936464408029242408285231434, −5.04156407802800391839279646075, −3.71940083293662842624512961666,
1.38465373844161639545120618531, 3.45211505394132585146673342538, 4.28390663474328467595816343746, 6.79734408292698575781176104086, 8.517856663854319407201648107735, 9.367576058966509396790587955239, 10.94583755299464704738208362473, 11.74852089541163803322684388626, 12.35483331107230776962059900515, 13.87807148142037748571719418610
