| L(s) = 1 | + (−1.40 + 0.170i)2-s + (−2.23 + 0.444i)3-s + (1.94 − 0.479i)4-s + (−2.33 − 3.49i)5-s + (3.06 − 1.00i)6-s + (−1.63 − 0.677i)7-s + (−2.64 + 1.00i)8-s + (2.03 − 0.842i)9-s + (3.86 + 4.50i)10-s + (−0.234 + 1.17i)11-s + (−4.13 + 1.93i)12-s + (−0.154 + 0.230i)13-s + (2.41 + 0.672i)14-s + (6.77 + 6.77i)15-s + (3.54 − 1.86i)16-s + (1.16 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.992 + 0.120i)2-s + (−1.29 + 0.256i)3-s + (0.970 − 0.239i)4-s + (−1.04 − 1.56i)5-s + (1.25 − 0.410i)6-s + (−0.618 − 0.256i)7-s + (−0.934 + 0.354i)8-s + (0.678 − 0.280i)9-s + (1.22 + 1.42i)10-s + (−0.0706 + 0.355i)11-s + (−1.19 + 0.558i)12-s + (−0.0428 + 0.0640i)13-s + (0.644 + 0.179i)14-s + (1.74 + 1.74i)15-s + (0.885 − 0.465i)16-s + (0.282 − 0.282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0732735 - 0.167700i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0732735 - 0.167700i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.170i)T \) |
| good | 3 | \( 1 + (2.23 - 0.444i)T + (2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (2.33 + 3.49i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (1.63 + 0.677i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.234 - 1.17i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (0.154 - 0.230i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 1.16i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.64 + 1.76i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (1.18 + 2.87i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.141 + 0.709i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 7.01iT - 31T^{2} \) |
| 37 | \( 1 + (-7.22 + 4.82i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (0.152 + 0.368i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (4.98 + 0.991i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 2.16i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.84 - 9.27i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-3.85 - 5.76i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (9.69 - 1.92i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-7.14 + 1.42i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (4.28 + 1.77i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.39 - 1.40i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.54 + 1.54i)T + 79iT^{2} \) |
| 83 | \( 1 + (9.93 + 6.63i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-3.81 + 9.20i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 9.29iT - 97T^{2} \) |
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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
−15.23510375471154552982849323431, −12.87022441895019348210387586890, −12.01775294377802892716546341607, −11.21446184025369803121978252926, −9.921547530637914418947397538768, −8.743906221717464486525720713791, −7.46712554997456361167029302364, −5.93137301791584047581122931212, −4.49667562310167096011446581900, −0.39444128040918195661110123005,
3.24196657506630518208294168169, 6.14173686388698147776386660345, 6.87541527993523366263021559474, 8.108204118362837161117167713785, 10.04804166709734884239738256615, 10.94217179152802888733063966185, 11.60860498100867990984245335397, 12.52423626760888171641595236983, 14.64851481301799998679265997777, 15.69189781488702087681732318194
