| L(s) = 1 | + (−1.50 − 1.18i)2-s + (−0.564 + 0.792i)3-s + (0.396 + 1.63i)4-s + (−0.812 + 0.156i)5-s + (1.79 − 0.526i)6-s + (−1.33 − 2.28i)7-s + (−0.253 + 0.554i)8-s + (0.671 + 1.94i)9-s + (1.41 + 0.727i)10-s + (−3.74 + 2.94i)11-s + (−1.52 − 0.608i)12-s + (−3.75 + 2.41i)13-s + (−0.688 + 5.03i)14-s + (0.334 − 0.732i)15-s + (4.03 − 2.07i)16-s + (4.06 + 3.87i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.838i)2-s + (−0.325 + 0.457i)3-s + (0.198 + 0.817i)4-s + (−0.363 + 0.0700i)5-s + (0.731 − 0.214i)6-s + (−0.505 − 0.862i)7-s + (−0.0894 + 0.195i)8-s + (0.223 + 0.646i)9-s + (0.446 + 0.230i)10-s + (−1.12 + 0.888i)11-s + (−0.438 − 0.175i)12-s + (−1.04 + 0.669i)13-s + (−0.184 + 1.34i)14-s + (0.0863 − 0.189i)15-s + (1.00 − 0.519i)16-s + (0.986 + 0.940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134388 + 0.160774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134388 + 0.160774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.33 + 2.28i)T \) |
| 23 | \( 1 + (4.76 + 0.569i)T \) |
| good | 2 | \( 1 + (1.50 + 1.18i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.564 - 0.792i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (0.812 - 0.156i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (3.74 - 2.94i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (3.75 - 2.41i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.06 - 3.87i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.735 + 0.701i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (6.21 - 1.82i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-9.48 + 0.905i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.19 + 9.23i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-1.65 - 1.91i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.75 + 3.83i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-1.59 + 2.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0701 + 1.47i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.25 + 0.646i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-3.59 - 5.04i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-0.641 + 0.256i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.382 + 2.65i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.48 - 14.3i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.428 - 8.99i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.874 + 1.00i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-9.14 - 0.872i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-2.97 - 3.43i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.78465505982718584968929705735, −11.88398371175339884826873502445, −10.78815826103341232979571788747, −10.08604476205786934468093878813, −9.694011055127124032125863108122, −7.991181612172926085506579763720, −7.36091114798159858837211087006, −5.38920733770409123735736204866, −4.00387396687050347475728501831, −2.17556727431797961745889276513,
0.27691127081737558956578050787, 3.12463736017775270383203340264, 5.47728887601823494634803495798, 6.33898317807467125044176465754, 7.63273086371976159119890259690, 8.145610849529434503241163268017, 9.511486221509587833970891671678, 10.10016433123245819428626494249, 11.84391089271309400579178396065, 12.35961875432616687131711209135
