L(s) = 1 | + (0.221 − 1.71i)3-s + (1.84 + 3.19i)5-s + (−0.926 + 2.47i)7-s + (−2.90 − 0.760i)9-s + (−0.446 + 0.772i)11-s + (0.598 − 1.03i)13-s + (5.90 − 2.46i)15-s + (−0.124 − 0.216i)17-s + (−1.40 + 2.43i)19-s + (4.05 + 2.14i)21-s + (1.23 + 2.14i)23-s + (−4.31 + 7.47i)25-s + (−1.94 + 4.81i)27-s + (2.07 + 3.58i)29-s − 3.58·31-s + ⋯ |
L(s) = 1 | + (0.127 − 0.991i)3-s + (0.825 + 1.43i)5-s + (−0.350 + 0.936i)7-s + (−0.967 − 0.253i)9-s + (−0.134 + 0.233i)11-s + (0.165 − 0.287i)13-s + (1.52 − 0.636i)15-s + (−0.0303 − 0.0525i)17-s + (−0.322 + 0.557i)19-s + (0.884 + 0.467i)21-s + (0.258 + 0.447i)23-s + (−0.863 + 1.49i)25-s + (−0.374 + 0.927i)27-s + (0.384 + 0.666i)29-s − 0.643·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.535525197\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535525197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.221 + 1.71i)T \) |
| 7 | \( 1 + (0.926 - 2.47i)T \) |
good | 5 | \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.446 - 0.772i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.124 + 0.216i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.07 - 3.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.98 - 8.64i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + (4.94 + 8.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.02T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 + (6.16 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.10613034173045721779826003233, −9.318558044346840042123077999161, −8.399725791219529256476938111144, −7.44932184920824980021320312166, −6.64600939545751716730388530256, −6.09206626922651715644330360287, −5.32185889790307047961925123231, −3.35611019471545829189951929903, −2.65998822119982972169648103332, −1.75448173981580667955203951564,
0.67432416799957010167286783723, 2.30050884823586810539645575230, 3.78726887476274525379073021608, 4.49690775582698544181416234538, 5.33699547899958000274851356759, 6.12781368440311092597474358662, 7.36040091248780279350114450587, 8.599019909531672816658420531833, 8.970035305026580709816571511881, 9.760459868460895196935469954817