L(s) = 1 | + (0.828 + 1.14i)2-s + 0.692·3-s + (−0.627 + 1.89i)4-s + (−0.245 − 2.22i)5-s + (0.573 + 0.794i)6-s + (−0.343 + 0.343i)7-s + (−2.69 + 0.853i)8-s − 2.52·9-s + (2.34 − 2.12i)10-s + (0.843 + 0.843i)11-s + (−0.434 + 1.31i)12-s − 3.68i·13-s + (−0.678 − 0.109i)14-s + (−0.169 − 1.53i)15-s + (−3.21 − 2.38i)16-s + (0.412 − 0.412i)17-s + ⋯ |
L(s) = 1 | + (0.585 + 0.810i)2-s + 0.399·3-s + (−0.313 + 0.949i)4-s + (−0.109 − 0.993i)5-s + (0.234 + 0.324i)6-s + (−0.129 + 0.129i)7-s + (−0.953 + 0.301i)8-s − 0.840·9-s + (0.741 − 0.671i)10-s + (0.254 + 0.254i)11-s + (−0.125 + 0.379i)12-s − 1.02i·13-s + (−0.181 − 0.0292i)14-s + (−0.0438 − 0.397i)15-s + (−0.802 − 0.596i)16-s + (0.0999 − 0.0999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12642 + 0.531088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12642 + 0.531088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.828 - 1.14i)T \) |
| 5 | \( 1 + (0.245 + 2.22i)T \) |
good | 3 | \( 1 - 0.692T + 3T^{2} \) |
| 7 | \( 1 + (0.343 - 0.343i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.843 - 0.843i)T + 11iT^{2} \) |
| 13 | \( 1 + 3.68iT - 13T^{2} \) |
| 17 | \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.37 - 5.37i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.08 + 3.08i)T + 23iT^{2} \) |
| 29 | \( 1 + (-4.22 + 4.22i)T - 29iT^{2} \) |
| 31 | \( 1 - 8.75iT - 31T^{2} \) |
| 37 | \( 1 - 5.41iT - 37T^{2} \) |
| 41 | \( 1 - 2.54iT - 41T^{2} \) |
| 43 | \( 1 - 4.30iT - 43T^{2} \) |
| 47 | \( 1 + (4.56 + 4.56i)T + 47iT^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + (-7.33 + 7.33i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.81 + 4.81i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + (-6.87 + 6.87i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 + 1.10T + 89T^{2} \) |
| 97 | \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \) |
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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
−14.46147509618014767302024170241, −13.68535160251445585980766102728, −12.51742016567534863662541002998, −11.83695301605717708495139802737, −9.778974757077144884671762554900, −8.500561779656269827049156071074, −7.86568137367942978447336439731, −6.06126074730888645115021594826, −4.96114928518326551801356461087, −3.31095402654251691635767114458,
2.59868002601957994118294373455, 3.84870330883114885208135802640, 5.72701751667994759984488845038, 7.09516803332818603083463163024, 8.913091231076844986459612065712, 9.963135954222803074020498021366, 11.31551292775281328764642806538, 11.73326463381085822762382503533, 13.47292556271921233357855621167, 14.08079406694190481363832209785