| L(s) = 1 | + (0.882 − 0.155i)2-s + (1.65 + 0.495i)3-s + (−1.12 + 0.409i)4-s + (0.123 − 0.0449i)5-s + (1.54 + 0.178i)6-s + (1.69 + 2.02i)7-s + (−2.48 + 1.43i)8-s + (2.50 + 1.64i)9-s + (0.101 − 0.0588i)10-s + (1.37 − 3.76i)11-s + (−2.07 + 0.122i)12-s + (−1.02 − 2.81i)13-s + (1.81 + 1.52i)14-s + (0.227 − 0.0134i)15-s + (−0.130 + 0.109i)16-s + (−0.172 − 0.298i)17-s + ⋯ |
| L(s) = 1 | + (0.623 − 0.109i)2-s + (0.958 + 0.286i)3-s + (−0.562 + 0.204i)4-s + (0.0552 − 0.0201i)5-s + (0.629 + 0.0730i)6-s + (0.642 + 0.766i)7-s + (−0.877 + 0.506i)8-s + (0.836 + 0.548i)9-s + (0.0322 − 0.0186i)10-s + (0.413 − 1.13i)11-s + (−0.597 + 0.0352i)12-s + (−0.284 − 0.781i)13-s + (0.484 + 0.407i)14-s + (0.0586 − 0.00346i)15-s + (−0.0326 + 0.0274i)16-s + (−0.0417 − 0.0723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81773 + 0.331707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81773 + 0.331707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.495i)T \) |
| 7 | \( 1 + (-1.69 - 2.02i)T \) |
| good | 2 | \( 1 + (-0.882 + 0.155i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.123 + 0.0449i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.37 + 3.76i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.02 + 2.81i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.172 + 0.298i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 + 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.27 + 1.10i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.29 + 6.31i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.98 - 8.20i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + 7.28T + 37T^{2} \) |
| 41 | \( 1 + (-9.04 + 3.29i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.350 - 1.98i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.17 - 1.52i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (5.49 + 3.17i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.136 - 0.114i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.59 - 4.38i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 11.3i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.373 - 0.215i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.52iT - 73T^{2} \) |
| 79 | \( 1 + (-0.410 - 2.32i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.81 - 2.48i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.70 - 13.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.2 + 2.15i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.75596778966437475044247450397, −11.91611879197219430239546637895, −10.68385493131811352453651422036, −9.385337636452348743284339709056, −8.561210377502070584248430164645, −7.967816612220842834433022035293, −6.01885687752001932535868617123, −4.87111013009300939021522093560, −3.72960770245025039523996421199, −2.51112055049641946326149327704,
1.91539112346061206793574047575, 3.99103022789796773953379393674, 4.47058159591585204609627421673, 6.27045005929313687875602899682, 7.39178715250901920376642841058, 8.437321950322206136796705163106, 9.539599455191411100001358005321, 10.25519080215104073173069728639, 11.95950451349650464757620339481, 12.68739518638277663301033313682
