| L(s) = 1 | + (−1.19 − 0.753i)2-s + (−1.51 + 1.01i)3-s + (0.865 + 1.80i)4-s + (−0.618 + 3.10i)5-s + (2.58 − 0.0705i)6-s + (1.63 − 0.325i)7-s + (0.322 − 2.80i)8-s + (0.130 − 0.314i)9-s + (3.08 − 3.25i)10-s + (−2.21 + 3.31i)11-s + (−3.14 − 1.86i)12-s + (−1.46 − 1.46i)13-s + (−2.20 − 0.843i)14-s + (−2.21 − 5.35i)15-s + (−2.50 + 3.11i)16-s + (3.44 − 2.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.846 − 0.532i)2-s + (−0.877 + 0.586i)3-s + (0.432 + 0.901i)4-s + (−0.276 + 1.39i)5-s + (1.05 − 0.0288i)6-s + (0.619 − 0.123i)7-s + (0.114 − 0.993i)8-s + (0.0433 − 0.104i)9-s + (0.974 − 1.02i)10-s + (−0.667 + 0.999i)11-s + (−0.908 − 0.537i)12-s + (−0.406 − 0.406i)13-s + (−0.589 − 0.225i)14-s + (−0.572 − 1.38i)15-s + (−0.625 + 0.779i)16-s + (0.835 − 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377136 + 0.274064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377136 + 0.274064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 + 0.753i)T \) |
| 17 | \( 1 + (-3.44 + 2.26i)T \) |
| good | 3 | \( 1 + (1.51 - 1.01i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (0.618 - 3.10i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 0.325i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (2.21 - 3.31i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (1.46 + 1.46i)T + 13iT^{2} \) |
| 19 | \( 1 + (-5.50 + 2.28i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (0.448 + 0.299i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-3.34 - 0.664i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-5.34 - 7.99i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-1.82 - 2.72i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.27 + 6.40i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.07 + 1.27i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (1.17 - 1.17i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.80 - 6.77i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.92 + 11.8i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.19 + 0.435i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 + (1.19 - 0.800i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 3.55i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (3.43 - 5.14i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.913 - 2.20i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-8.54 + 8.54i)T - 89iT^{2} \) |
| 97 | \( 1 + (6.56 + 1.30i)T + (89.6 + 37.1i)T^{2} \) |
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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
−15.35158395869901826889310283859, −14.02213622144046368033452004826, −12.18524557988934901380493389490, −11.38993773345263664994656772352, −10.45665268931580677110119702297, −9.927929413501457848159266873324, −7.88305111685602659633169848789, −6.98155072920108397635672599916, −4.96404710458957712958575682121, −2.91991761865817571294090917683,
1.01531978256073973867723635746, 5.11200062035260273514543586479, 5.98885444359288309468935012454, 7.71793242533270531961646809093, 8.494703259342992637961901004640, 9.862398648095641538452748432407, 11.44443058948674099654024008295, 12.02204851510051786205897314583, 13.40576828850687953158704556168, 14.79833392559037893397710282155
