L(s) = 1 | + (−2.19 − 1.26i)3-s + (2.31 + 1.28i)7-s + (1.70 + 2.95i)9-s + (2.11 − 3.66i)11-s + 5.98i·13-s + (−6.61 − 3.81i)17-s + (3.47 + 6.01i)19-s + (−3.43 − 5.74i)21-s + (−1.48 + 0.858i)23-s − 1.04i·27-s − 5.18·29-s + (0.254 − 0.441i)31-s + (−9.26 + 5.35i)33-s + (−3.45 + 1.99i)37-s + (7.57 − 13.1i)39-s + ⋯ |
L(s) = 1 | + (−1.26 − 0.730i)3-s + (0.873 + 0.486i)7-s + (0.568 + 0.984i)9-s + (0.637 − 1.10i)11-s + 1.66i·13-s + (−1.60 − 0.926i)17-s + (0.796 + 1.37i)19-s + (−0.750 − 1.25i)21-s + (−0.310 + 0.178i)23-s − 0.200i·27-s − 0.962·29-s + (0.0457 − 0.0792i)31-s + (−1.61 + 0.931i)33-s + (−0.567 + 0.327i)37-s + (1.21 − 2.10i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8679043250\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8679043250\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.31 - 1.28i)T \) |
good | 3 | \( 1 + (2.19 + 1.26i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.98iT - 13T^{2} \) |
| 17 | \( 1 + (6.61 + 3.81i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.47 - 6.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.48 - 0.858i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 + (-0.254 + 0.441i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.45 - 1.99i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 - 4.17iT - 43T^{2} \) |
| 47 | \( 1 + (1.64 - 0.946i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.53 + 3.19i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.79 - 4.84i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.4 - 6.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + (-4.66 - 2.69i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.673 + 1.16i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.70iT - 83T^{2} \) |
| 89 | \( 1 + (-5.42 - 9.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.6iT - 97T^{2} \) |
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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
−9.542650205497111540110491440957, −8.913266011718395220785762742535, −7.979093354727160850841333676165, −7.01239186930845929358193217384, −6.38294104177584289884417961399, −5.66994139668027244070517244338, −4.84625757945404040530128752414, −3.83949074684995662456111361133, −2.13037481172763619799067264912, −1.21024445792985370770897925316,
0.46411040280448948931566954867, 2.01108845809719222690442437510, 3.68891153508379474923760051894, 4.63409070599395247960738247520, 5.04455836890390554663383195250, 6.01575316488239418647463059948, 6.91538659589435379912691921803, 7.70815425966901726604320574897, 8.745844479064192682986224524543, 9.669312334843617106767454096965