L(s) = 1 | + (−1.28 − 1.16i)3-s + (−1.93 + 1.80i)7-s + (0.284 + 2.98i)9-s + (−4.05 + 2.34i)11-s − 2.18·13-s + (−6.49 + 3.74i)17-s + (0.638 + 0.368i)19-s + (4.58 − 0.0477i)21-s + (4.03 − 6.99i)23-s + (3.11 − 4.15i)27-s + 1.15i·29-s + (8.95 − 5.16i)31-s + (7.92 + 1.72i)33-s + (−3.99 − 2.30i)37-s + (2.80 + 2.55i)39-s + ⋯ |
L(s) = 1 | + (−0.739 − 0.672i)3-s + (−0.732 + 0.680i)7-s + (0.0947 + 0.995i)9-s + (−1.22 + 0.706i)11-s − 0.607·13-s + (−1.57 + 0.909i)17-s + (0.146 + 0.0845i)19-s + (0.999 − 0.0104i)21-s + (0.842 − 1.45i)23-s + (0.599 − 0.800i)27-s + 0.214i·29-s + (1.60 − 0.928i)31-s + (1.37 + 0.300i)33-s + (−0.657 − 0.379i)37-s + (0.449 + 0.408i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6297196191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6297196191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.28 + 1.16i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.93 - 1.80i)T \) |
good | 11 | \( 1 + (4.05 - 2.34i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + (6.49 - 3.74i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.638 - 0.368i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.03 + 6.99i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.15iT - 29T^{2} \) |
| 31 | \( 1 + (-8.95 + 5.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.99 + 2.30i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 + 9.24iT - 43T^{2} \) |
| 47 | \( 1 + (-7.52 - 4.34i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.06 + 7.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.48 - 6.04i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.13 - 2.96i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.19 + 0.691i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.26iT - 71T^{2} \) |
| 73 | \( 1 + (0.122 + 0.211i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.79 + 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.4iT - 83T^{2} \) |
| 89 | \( 1 + (-0.658 + 1.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−8.795200936134755928522860836081, −8.193427792869443849107542832430, −7.15769786055162041616266082641, −6.66647601068632661864021243104, −5.85297429986981305698318390685, −5.04991193064730801555208336443, −4.29053421464038743017434354890, −2.59435454259541587580980331749, −2.21199883099715860635048155701, −0.35687946430654700392942391740,
0.74920879589748775078249305392, 2.71639165714997757659380458161, 3.43922876103140442628178665558, 4.63613227561660664584634255931, 5.08300822432711302975536178343, 6.09749370596704801597210458044, 6.84648359557201923392639672729, 7.52512231124252907134697894304, 8.648146179129165066847417457287, 9.460281278429575403376978645917