L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + (3.08 + 1.78i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (2.06 + 3.57i)11-s − 0.999i·12-s + (−1.35 − 3.34i)13-s − 3.56·14-s + (−0.5 − 0.866i)16-s + (4.43 + 2.56i)17-s + 0.999i·18-s + (−1.78 + 3.08i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + (1.16 + 0.673i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.621 + 1.07i)11-s − 0.288i·12-s + (−0.375 − 0.926i)13-s − 0.951·14-s + (−0.125 − 0.216i)16-s + (1.07 + 0.621i)17-s + 0.235i·18-s + (−0.408 + 0.707i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780160217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780160217\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1.35 + 3.34i)T \) |
good | 7 | \( 1 + (-3.08 - 1.78i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.43 - 2.56i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 - 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.65 - 3.84i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.28 - 5.68i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + (3.57 - 2.06i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 3.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.95 + 2.28i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 4.43iT - 53T^{2} \) |
| 59 | \( 1 + (-5.28 + 9.14i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.3 - 7.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.43 + 4.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 - 1.12iT - 83T^{2} \) |
| 89 | \( 1 + (-0.903 - 1.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.972 - 0.561i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.162397805237674859988866439110, −8.299607109297916221437325260708, −7.978254355870401287363924611261, −7.25642776847261496948891801172, −6.22572631363953724593789714273, −5.43609571278254169444807948418, −4.55661439048021856151043230803, −3.34632445042360923801495605191, −2.03668451704657700972991029318, −1.41732411732129025527618781081,
0.797729931412272096327385139835, 1.94957781136048794268062162836, 2.98872391811124551503370218122, 4.13213283209091079438763665534, 4.61414282823250077969365799517, 5.95399370023570637189759726150, 6.92348248093424087718847340380, 7.76893276126941448376774295174, 8.355723057183330982725833495188, 8.950697408882482234907572345400