L(s) = 1 | + 1.90i·2-s − i·3-s − 1.62·4-s + (0.311 + 2.21i)5-s + 1.90·6-s + i·7-s + 0.719i·8-s − 9-s + (−4.21 + 0.592i)10-s + 2·11-s + 1.62i·12-s − 6.42i·13-s − 1.90·14-s + (2.21 − 0.311i)15-s − 4.61·16-s − 4.42i·17-s + ⋯ |
L(s) = 1 | + 1.34i·2-s − 0.577i·3-s − 0.811·4-s + (0.139 + 0.990i)5-s + 0.776·6-s + 0.377i·7-s + 0.254i·8-s − 0.333·9-s + (−1.33 + 0.187i)10-s + 0.603·11-s + 0.468i·12-s − 1.78i·13-s − 0.508·14-s + (0.571 − 0.0803i)15-s − 1.15·16-s − 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.691672 + 0.795645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.691672 + 0.795645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.311 - 2.21i)T \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 - 1.90iT - 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6.42iT - 13T^{2} \) |
| 17 | \( 1 + 4.42iT - 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 2.75iT - 47T^{2} \) |
| 53 | \( 1 - 9.18iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 1.57iT - 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 11.9iT - 97T^{2} \) |
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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
−14.23674687897973100047271979737, −13.45829095224240761971630202673, −12.03241079184143613288169175123, −10.95657576744041577766152525330, −9.504637956116825521260435185910, −8.084595345053140324737395304580, −7.30837504742868795446581709238, −6.29451319876752336862472580399, −5.35059424126018785705391587969, −2.89337844562107903556268616422,
1.65761090081185543965036771449, 3.76884378217202253573176078288, 4.66538171949640035486101255138, 6.57151319130391681096079613420, 8.585556909406263491608478922031, 9.467634483453780020428146994259, 10.29807747160142773847567369142, 11.56346165138034547950714484030, 12.08270890643338032845338257683, 13.30769290291366378232357486059