| L(s) = 1 | + (0.406 + 1.35i)2-s + (1.06 − 0.714i)3-s + (−1.66 + 1.10i)4-s + (−0.330 + 0.0657i)5-s + (1.40 + 1.15i)6-s + (−0.739 − 1.78i)7-s + (−2.17 − 1.81i)8-s + (−0.515 + 1.24i)9-s + (−0.223 − 0.420i)10-s + (0.971 − 1.45i)11-s + (−0.996 + 2.37i)12-s + (−3.70 − 0.737i)13-s + (2.11 − 1.72i)14-s + (−0.306 + 0.306i)15-s + (1.56 − 3.67i)16-s + (4.47 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (0.287 + 0.957i)2-s + (0.617 − 0.412i)3-s + (−0.834 + 0.551i)4-s + (−0.147 + 0.0294i)5-s + (0.572 + 0.472i)6-s + (−0.279 − 0.674i)7-s + (−0.767 − 0.640i)8-s + (−0.171 + 0.414i)9-s + (−0.0706 − 0.133i)10-s + (0.292 − 0.438i)11-s + (−0.287 + 0.684i)12-s + (−1.02 − 0.204i)13-s + (0.565 − 0.461i)14-s + (−0.0791 + 0.0791i)15-s + (0.392 − 0.919i)16-s + (1.08 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945731 + 0.438123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945731 + 0.438123i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.406 - 1.35i)T \) |
| good | 3 | \( 1 + (-1.06 + 0.714i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (0.330 - 0.0657i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (0.739 + 1.78i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.971 + 1.45i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (3.70 + 0.737i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-4.47 - 4.47i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.16 + 5.83i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (1.28 + 0.531i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-3.04 - 4.56i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 + (0.910 + 4.57i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.66 - 1.10i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (5.83 + 3.89i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (0.0482 + 0.0482i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.43 + 9.62i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (2.89 - 0.576i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (0.675 - 0.451i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (2.41 - 1.61i)T + (25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (2.88 + 6.96i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-1.92 + 4.64i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 10.5i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.0104 - 0.0525i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (7.52 - 3.11i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−14.87466393982116150144140640608, −14.08800965072653909947064454785, −13.26712711307544273273733772878, −12.18533851598248091635688615105, −10.36392183230169066903447557639, −8.891248093632817503527668252862, −7.79585622022863202690669163766, −6.89150753712141902549524607451, −5.19126053499004120598640578927, −3.39761895470206424510107055296,
2.68156668465903861420581420937, 4.13560996150112831269217416245, 5.79815770037691418502390253319, 8.022481695355883291418557319201, 9.609532933087064328138085386156, 9.783457831136322448257733742675, 11.91112145880834854504386741732, 12.12299933186787989924304904520, 13.79452743407239179169161053381, 14.62630828387388154881219480278
