| L(s) = 1 | + (0.0702 + 0.997i)2-s + (1.38 + 0.262i)3-s + (−0.990 + 0.140i)4-s + (1.40 + 2.62i)5-s + (−0.164 + 1.39i)6-s + (3.20 + 0.764i)7-s + (−0.209 − 0.977i)8-s + (−0.951 − 0.374i)9-s + (−2.51 + 1.58i)10-s + (1.81 − 0.918i)11-s + (−1.40 − 0.0659i)12-s + (−0.642 + 0.405i)13-s + (−0.537 + 3.24i)14-s + (1.25 + 3.99i)15-s + (0.960 − 0.277i)16-s + (−1.43 − 0.724i)17-s + ⋯ |
| L(s) = 1 | + (0.0496 + 0.705i)2-s + (0.797 + 0.151i)3-s + (−0.495 + 0.0701i)4-s + (0.629 + 1.17i)5-s + (−0.0671 + 0.570i)6-s + (1.20 + 0.288i)7-s + (−0.0740 − 0.345i)8-s + (−0.317 − 0.124i)9-s + (−0.796 + 0.502i)10-s + (0.546 − 0.277i)11-s + (−0.405 − 0.0190i)12-s + (−0.178 + 0.112i)13-s + (−0.143 + 0.867i)14-s + (0.324 + 1.03i)15-s + (0.240 − 0.0694i)16-s + (−0.347 − 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42739 + 1.60638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42739 + 1.60638i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0702 - 0.997i)T \) |
| 269 | \( 1 + (-10.0 - 12.9i)T \) |
| good | 3 | \( 1 + (-1.38 - 0.262i)T + (2.79 + 1.09i)T^{2} \) |
| 5 | \( 1 + (-1.40 - 2.62i)T + (-2.76 + 4.16i)T^{2} \) |
| 7 | \( 1 + (-3.20 - 0.764i)T + (6.24 + 3.16i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 0.918i)T + (6.50 - 8.86i)T^{2} \) |
| 13 | \( 1 + (0.642 - 0.405i)T + (5.60 - 11.7i)T^{2} \) |
| 17 | \( 1 + (1.43 + 0.724i)T + (10.0 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.000188 + 0.00804i)T + (-18.9 - 0.890i)T^{2} \) |
| 23 | \( 1 + (-2.02 - 0.384i)T + (21.4 + 8.42i)T^{2} \) |
| 29 | \( 1 + (5.63 + 8.49i)T + (-11.2 + 26.7i)T^{2} \) |
| 31 | \( 1 + (0.417 - 0.568i)T + (-9.30 - 29.5i)T^{2} \) |
| 37 | \( 1 + (0.00525 - 0.00538i)T + (-0.867 - 36.9i)T^{2} \) |
| 41 | \( 1 + (0.556 - 7.90i)T + (-40.5 - 5.74i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 5.97i)T + (-16.6 + 39.6i)T^{2} \) |
| 47 | \( 1 + (3.57 + 4.42i)T + (-9.84 + 45.9i)T^{2} \) |
| 53 | \( 1 + (1.30 - 3.56i)T + (-40.4 - 34.2i)T^{2} \) |
| 59 | \( 1 + (1.38 - 0.195i)T + (56.6 - 16.3i)T^{2} \) |
| 61 | \( 1 + (-4.35 + 4.90i)T + (-7.13 - 60.5i)T^{2} \) |
| 67 | \( 1 + (-1.68 - 0.239i)T + (64.3 + 18.6i)T^{2} \) |
| 71 | \( 1 + (0.507 + 3.06i)T + (-67.2 + 22.8i)T^{2} \) |
| 73 | \( 1 + (-6.24 - 2.12i)T + (57.8 + 44.5i)T^{2} \) |
| 79 | \( 1 + (1.53 + 2.56i)T + (-37.3 + 69.6i)T^{2} \) |
| 83 | \( 1 + (4.39 + 4.09i)T + (5.83 + 82.7i)T^{2} \) |
| 89 | \( 1 + (4.10 + 15.5i)T + (-77.4 + 43.8i)T^{2} \) |
| 97 | \( 1 + (-4.08 - 4.59i)T + (-11.3 + 96.3i)T^{2} \) |
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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
−11.16862535747160176696894877305, −9.903386559097434671696854051679, −9.193040632250956574715031580772, −8.306705659626865049573182851059, −7.56535781325283350096359554195, −6.48950631284623813154433048925, −5.70949693445334167430551315821, −4.45200199904331949391475516914, −3.18647720189889933263904683739, −2.08304997783037481585523018891,
1.36316151361428011588022167869, 2.19895684125340067869421056243, 3.74443243090321511952502759907, 4.86471782466749605644576454313, 5.52077347757194306398988870546, 7.24103245705967536769882407170, 8.288751467066349220807838288015, 8.887083779669226731874795276456, 9.457556998375003008637786351962, 10.69762758077902887670899921929
