| L(s) = 1 | + (1.57 − 0.574i)2-s + (0.632 − 0.530i)4-s + (0.196 + 1.11i)5-s + (−2.99 − 2.51i)7-s + (−0.987 + 1.70i)8-s + (0.949 + 1.64i)10-s + (0.324 − 1.84i)11-s + (0.688 + 0.250i)13-s + (−6.17 − 2.24i)14-s + (−0.862 + 4.89i)16-s + (0.944 + 1.63i)17-s + (−1.37 + 2.37i)19-s + (0.714 + 0.599i)20-s + (−0.545 − 3.09i)22-s + (4.46 − 3.74i)23-s + ⋯ |
| L(s) = 1 | + (1.11 − 0.406i)2-s + (0.316 − 0.265i)4-s + (0.0877 + 0.497i)5-s + (−1.13 − 0.949i)7-s + (−0.349 + 0.604i)8-s + (0.300 + 0.519i)10-s + (0.0979 − 0.555i)11-s + (0.190 + 0.0694i)13-s + (−1.65 − 0.600i)14-s + (−0.215 + 1.22i)16-s + (0.229 + 0.396i)17-s + (−0.314 + 0.544i)19-s + (0.159 + 0.134i)20-s + (−0.116 − 0.660i)22-s + (0.930 − 0.781i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38628 - 0.233344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38628 - 0.233344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.57 + 0.574i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.196 - 1.11i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.99 + 2.51i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.324 + 1.84i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.688 - 0.250i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.46 + 3.74i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.99 - 1.81i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 0.861i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.69 + 2.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.68 - 0.614i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.873 + 4.95i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.30 + 1.09i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + (-1.95 - 11.0i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.00 - 3.36i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.77 - 0.646i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.09 + 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.6 - 4.22i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (10.9 - 3.99i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0596 - 0.338i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.10974799227658089772073767190, −13.19674492399947792358424316236, −12.52107236926894776608744011867, −11.12122653551985099949746611155, −10.27706914639364562360105001585, −8.729625656545588850245407848415, −6.98088290466313881922702040922, −5.84592701996927407872817611449, −4.09237583792113210719002491654, −3.05982508720670982471144215024,
3.16761777815462144155723855216, 4.83137412678742614477004440869, 5.89569942669989025626940443100, 7.02501178964556677212114939915, 8.987985031120237624713940020776, 9.735870766597500111039117182289, 11.62179024204801028589029035421, 12.86813358238099881722972175673, 13.01936866281577302566801372628, 14.46020590864643005662271034151
