| L(s) = 1 | + (−2.26 + 0.399i)2-s + (0.415 + 1.68i)3-s + (3.10 − 1.12i)4-s + (−1.73 + 0.630i)5-s + (−1.61 − 3.64i)6-s + (0.678 + 2.55i)7-s + (−2.59 + 1.49i)8-s + (−2.65 + 1.39i)9-s + (3.67 − 2.12i)10-s + (−0.0420 + 0.115i)11-s + (3.18 + 4.74i)12-s + (−0.0159 − 0.0437i)13-s + (−2.56 − 5.52i)14-s + (−1.78 − 2.65i)15-s + (0.221 − 0.185i)16-s + (0.0786 + 0.136i)17-s + ⋯ |
| L(s) = 1 | + (−1.60 + 0.282i)2-s + (0.239 + 0.970i)3-s + (1.55 − 0.564i)4-s + (−0.774 + 0.281i)5-s + (−0.659 − 1.48i)6-s + (0.256 + 0.966i)7-s + (−0.916 + 0.528i)8-s + (−0.884 + 0.465i)9-s + (1.16 − 0.670i)10-s + (−0.0126 + 0.0347i)11-s + (0.919 + 1.36i)12-s + (−0.00441 − 0.0121i)13-s + (−0.684 − 1.47i)14-s + (−0.459 − 0.684i)15-s + (0.0553 − 0.0464i)16-s + (0.0190 + 0.0330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00895971 + 0.357059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00895971 + 0.357059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 - 1.68i)T \) |
| 7 | \( 1 + (-0.678 - 2.55i)T \) |
| good | 2 | \( 1 + (2.26 - 0.399i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (1.73 - 0.630i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.0420 - 0.115i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0159 + 0.0437i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.0786 - 0.136i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.57 + 3.79i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.957 + 0.168i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.33 - 9.16i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.56 + 4.29i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 + (-5.76 + 2.09i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.67 - 9.52i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.91 - 1.42i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.141 + 0.0818i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.06 - 7.60i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.03 + 2.85i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.70 - 9.68i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.863 + 0.498i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.18iT - 73T^{2} \) |
| 79 | \( 1 + (-0.451 - 2.56i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (6.02 + 2.19i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.30 - 9.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.2 + 2.33i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.84751082785057354092257919304, −11.26662482750670937095653268435, −11.08294089713390026023614091063, −9.819687190530990834470020588806, −8.970272712869335737370807478549, −8.353070658235065910458964125779, −7.34874273007926857798116508130, −5.91191922019545921426005175137, −4.29211709717953408613137231544, −2.50605915936198464399707557294,
0.50023944646759201840991336026, 2.07047001444578576329152984605, 4.02384284149839830123277446202, 6.33011610290701911705004853177, 7.55362328962896513869155627986, 7.955256060352991584567526704516, 8.828222437842019949909162029684, 10.05799064967584708993446849710, 11.04175934160175624469197340417, 11.80643317784817093375561974141
