L(s) = 1 | − 1.17i·2-s + 0.539·3-s + 0.630·4-s + 0.460i·5-s − 0.630i·6-s + i·7-s − 3.07i·8-s − 2.70·9-s + 0.539·10-s − 0.829i·11-s + 0.340·12-s + (−2.87 + 2.17i)13-s + 1.17·14-s + 0.248i·15-s − 2.34·16-s + 2.87·17-s + ⋯ |
L(s) = 1 | − 0.827i·2-s + 0.311·3-s + 0.315·4-s + 0.206i·5-s − 0.257i·6-s + 0.377i·7-s − 1.08i·8-s − 0.903·9-s + 0.170·10-s − 0.250i·11-s + 0.0981·12-s + (−0.798 + 0.601i)13-s + 0.312·14-s + 0.0641i·15-s − 0.585·16-s + 0.698·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01036 - 0.503701i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01036 - 0.503701i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 13 | \( 1 + (2.87 - 2.17i)T \) |
good | 2 | \( 1 + 1.17iT - 2T^{2} \) |
| 3 | \( 1 - 0.539T + 3T^{2} \) |
| 5 | \( 1 - 0.460iT - 5T^{2} \) |
| 11 | \( 1 + 0.829iT - 11T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.32iT - 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 - 0.261T + 29T^{2} \) |
| 31 | \( 1 - 6.80iT - 31T^{2} \) |
| 37 | \( 1 + 9.51iT - 37T^{2} \) |
| 41 | \( 1 + 6.68iT - 41T^{2} \) |
| 43 | \( 1 - 0.418T + 43T^{2} \) |
| 47 | \( 1 + 9.24iT - 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 - 2.78iT - 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 5.07iT - 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + 0.353iT - 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 5.43iT - 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−14.05068252754954068192883691884, −12.45523679374304928334475730042, −11.90485926727221512169185059426, −10.78490521447773331975272410428, −9.780408507737456583356864958378, −8.535958805173087860680457716634, −7.11616696485486909006267447342, −5.68127986712893890694266321547, −3.58782396749804202945136524624, −2.27275681099378815718370678345,
2.76963395708607092435885909491, 4.96348232972901228501914286517, 6.22676806309552780090629189307, 7.53539649918199473425592615145, 8.331302842817510185479288024470, 9.745783686367274712587899073540, 11.09276162873804858964759355944, 12.11445807734131386658865890524, 13.47806499756116549839170074580, 14.55489268704814568117068852782