L(s) = 1 | + (−0.932 + 1.06i)2-s + (−0.260 − 1.98i)4-s + 0.546i·5-s − i·7-s + (2.35 + 1.57i)8-s + (−0.580 − 0.509i)10-s + 4.32·11-s − 3.39·13-s + (1.06 + 0.932i)14-s + (−3.86 + 1.03i)16-s + 0.0960i·17-s − 7.78i·19-s + (1.08 − 0.142i)20-s + (−4.03 + 4.60i)22-s − 0.716·23-s + ⋯ |
L(s) = 1 | + (−0.659 + 0.751i)2-s + (−0.130 − 0.991i)4-s + 0.244i·5-s − 0.377i·7-s + (0.831 + 0.555i)8-s + (−0.183 − 0.161i)10-s + 1.30·11-s − 0.940·13-s + (0.284 + 0.249i)14-s + (−0.965 + 0.258i)16-s + 0.0232i·17-s − 1.78i·19-s + (0.242 − 0.0318i)20-s + (−0.860 + 0.981i)22-s − 0.149·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08011 + 0.0707370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08011 + 0.0707370i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.932 - 1.06i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.546iT - 5T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 0.0960iT - 17T^{2} \) |
| 19 | \( 1 + 7.78iT - 19T^{2} \) |
| 23 | \( 1 + 0.716T + 23T^{2} \) |
| 29 | \( 1 - 2.80iT - 29T^{2} \) |
| 31 | \( 1 + 1.38iT - 31T^{2} \) |
| 37 | \( 1 - 9.87T + 37T^{2} \) |
| 41 | \( 1 + 9.42iT - 41T^{2} \) |
| 43 | \( 1 + 1.12iT - 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 - 0.760T + 73T^{2} \) |
| 79 | \( 1 - 1.95iT - 79T^{2} \) |
| 83 | \( 1 + 3.56T + 83T^{2} \) |
| 89 | \( 1 + 5.30iT - 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−10.23387073710286263798123118430, −9.205646763460630787987551149770, −8.907122352145864645261393218552, −7.52936255552973472643421755485, −7.02640902519516167820408608151, −6.25031515394220926578847500569, −5.05771134648032462857547461902, −4.16157120540409575138245184075, −2.48185503984788506930688600138, −0.842048017810276841585004825465,
1.20736500466952216717766231463, 2.45168094020583427101313856276, 3.70540659087173432538959216311, 4.60456680985959461545809494910, 5.97895596819721626845049503724, 7.03740742834591636465338745467, 8.012854305932929576473759401419, 8.734459351357935307854043202913, 9.647132144769662196120080536793, 10.06357828816311020478211965864