| L(s) = 1 | + (0.868 + 2.01i)2-s + (−0.540 + 1.64i)3-s + (−1.92 + 2.04i)4-s + (0.209 − 3.59i)5-s + (−3.78 + 0.341i)6-s + (1.42 − 0.337i)7-s + (−1.66 − 0.606i)8-s + (−2.41 − 1.77i)9-s + (7.41 − 2.69i)10-s + (−3.76 + 2.47i)11-s + (−2.32 − 4.27i)12-s + (3.05 − 0.356i)13-s + (1.91 + 2.57i)14-s + (5.80 + 2.28i)15-s + (0.100 + 1.72i)16-s + (−2.57 − 2.15i)17-s + ⋯ |
| L(s) = 1 | + (0.614 + 1.42i)2-s + (−0.311 + 0.950i)3-s + (−0.963 + 1.02i)4-s + (0.0936 − 1.60i)5-s + (−1.54 + 0.139i)6-s + (0.537 − 0.127i)7-s + (−0.589 − 0.214i)8-s + (−0.805 − 0.592i)9-s + (2.34 − 0.853i)10-s + (−1.13 + 0.747i)11-s + (−0.669 − 1.23i)12-s + (0.846 − 0.0989i)13-s + (0.511 + 0.687i)14-s + (1.49 + 0.590i)15-s + (0.0251 + 0.431i)16-s + (−0.624 − 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.646033 + 0.953428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.646033 + 0.953428i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.540 - 1.64i)T \) |
| good | 2 | \( 1 + (-0.868 - 2.01i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.209 + 3.59i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-1.42 + 0.337i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (3.76 - 2.47i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-3.05 + 0.356i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (2.57 + 2.15i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.625 + 0.525i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (2.81 + 0.666i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-5.35 + 7.19i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (0.183 + 0.611i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.652 - 3.69i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (3.86 - 8.95i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-0.0539 - 0.0271i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (0.628 - 2.09i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (0.875 - 1.51i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.24 - 5.42i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.91 - 2.03i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (0.188 + 0.253i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-0.0557 + 0.0202i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.24 + 0.453i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.61 + 8.38i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (5.47 + 12.6i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (-13.7 - 4.99i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.652 + 11.1i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−15.06598590445548199946746479223, −13.73905449318025526083754918015, −12.97513060974248021175127608879, −11.63287870018463883976019650879, −10.07952370974394180697727713656, −8.717629453070793590836129873177, −7.933256562885972396040541332397, −6.07020938375161072283508321822, −4.92644543067609081081691176828, −4.47398292548583394756206948287,
2.13035887544322534160090050264, 3.34740554704191043553843569533, 5.49971119254136021408421238421, 6.85823170621668810931313692229, 8.280608061985726388230547328385, 10.49979951050587432909018015551, 10.88976043218969983617622157092, 11.70482868383554949987903889441, 12.91361822955400440367583699063, 13.80471582231396735260771939957
