| L(s) = 1 | + (3.24 + 0.513i)2-s + (1.43 + 2.81i)3-s + (6.46 + 2.09i)4-s + (3.20 + 9.85i)6-s + (−7.51 − 7.51i)7-s + (8.17 + 4.16i)8-s + (−0.557 + 0.767i)9-s + (−1.41 + 1.03i)11-s + (3.35 + 21.1i)12-s + (−4.03 + 0.639i)13-s + (−20.5 − 28.2i)14-s + (2.40 + 1.75i)16-s + (−4.63 + 9.09i)17-s + (−2.20 + 2.20i)18-s + (23.1 − 7.51i)19-s + ⋯ |
| L(s) = 1 | + (1.62 + 0.256i)2-s + (0.477 + 0.936i)3-s + (1.61 + 0.524i)4-s + (0.533 + 1.64i)6-s + (−1.07 − 1.07i)7-s + (1.02 + 0.520i)8-s + (−0.0619 + 0.0852i)9-s + (−0.129 + 0.0937i)11-s + (0.279 + 1.76i)12-s + (−0.310 + 0.0491i)13-s + (−1.46 − 2.01i)14-s + (0.150 + 0.109i)16-s + (−0.272 + 0.535i)17-s + (−0.122 + 0.122i)18-s + (1.21 − 0.395i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.01759 + 1.34993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.01759 + 1.34993i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-3.24 - 0.513i)T + (3.80 + 1.23i)T^{2} \) |
| 3 | \( 1 + (-1.43 - 2.81i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (7.51 + 7.51i)T + 49iT^{2} \) |
| 11 | \( 1 + (1.41 - 1.03i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (4.03 - 0.639i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (4.63 - 9.09i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-23.1 + 7.51i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (3.93 - 24.8i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (0.252 + 0.0821i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 3.40i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (3.97 + 25.0i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (1.53 + 1.11i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (34.0 - 34.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (41.0 - 20.9i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-1.37 - 2.70i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-11.7 + 16.1i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-63.6 + 46.2i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (36.9 - 72.5i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (0.716 - 2.20i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-15.7 + 99.1i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (16.4 + 5.33i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (3.99 + 2.03i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-85.4 - 117. i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-124. + 63.3i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−13.43409498012156667694268238398, −12.69624679857401320742208673484, −11.44787623546422820480933588274, −10.16594825399712621974256269112, −9.368093290766647263793902693917, −7.43951235956932362654160290711, −6.44209010157791141093726076540, −5.01066251622807202935489875690, −3.88007304339198529306832521046, −3.18855632984424298410016670319,
2.30710284482682718871236019051, 3.23221897973524891808181844804, 5.02714636338341621302508139953, 6.17755676637729042775151410759, 7.11717868784559355689674710676, 8.622105828036419423185853998779, 10.03217877210907874653163052699, 11.67481445293916853793971614283, 12.33055182150737142351714613494, 13.05873985695584740727943760239
