L(s) = 1 | + (0.685 + 0.222i)3-s + (−2.21 + 0.277i)5-s + 4.42i·7-s + (−2.00 − 1.45i)9-s + (−3.36 + 2.44i)11-s + (0.592 − 0.815i)13-s + (−1.58 − 0.304i)15-s + (−3.49 + 1.13i)17-s + (−0.865 − 2.66i)19-s + (−0.985 + 3.03i)21-s + (4.41 + 6.08i)23-s + (4.84 − 1.22i)25-s + (−2.32 − 3.19i)27-s + (−2.81 + 8.66i)29-s + (0.593 + 1.82i)31-s + ⋯ |
L(s) = 1 | + (0.395 + 0.128i)3-s + (−0.992 + 0.123i)5-s + 1.67i·7-s + (−0.668 − 0.485i)9-s + (−1.01 + 0.737i)11-s + (0.164 − 0.226i)13-s + (−0.408 − 0.0785i)15-s + (−0.848 + 0.275i)17-s + (−0.198 − 0.610i)19-s + (−0.215 + 0.661i)21-s + (0.921 + 1.26i)23-s + (0.969 − 0.245i)25-s + (−0.446 − 0.615i)27-s + (−0.522 + 1.60i)29-s + (0.106 + 0.328i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.339183 + 0.718185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.339183 + 0.718185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.21 - 0.277i)T \) |
good | 3 | \( 1 + (-0.685 - 0.222i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.42iT - 7T^{2} \) |
| 11 | \( 1 + (3.36 - 2.44i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.592 + 0.815i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.49 - 1.13i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.865 + 2.66i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 6.08i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.81 - 8.66i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.593 - 1.82i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.77 + 7.95i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.62 - 4.81i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.12iT - 43T^{2} \) |
| 47 | \( 1 + (4.17 + 1.35i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.585 + 0.190i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.72 + 1.98i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.75 + 4.18i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.37 + 0.447i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.05 - 3.23i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.838 - 1.15i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.38 - 13.5i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.70 - 0.552i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.97 + 7.24i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.35 - 1.41i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.44009347193105619669940718868, −11.02707366043426568146520877997, −9.465664698630404275024256965795, −8.850869376033382125042040433434, −8.088590506336861871226485457187, −7.04191553163564514704684904657, −5.75048983547324883170326211288, −4.81708838114542188888182919048, −3.31742173597990731053162789600, −2.44580242967457421102962236818,
0.47576088814830292337281841183, 2.72077132395017895541507224411, 3.92634389165901819725351205734, 4.79524023821691961349572825473, 6.34196880474849359261109756121, 7.53955278325274613334677398449, 7.987118467872972238647956405481, 8.865854403732556950912942175330, 10.30895815778117831871700504035, 10.96871390942405338892990018469