L(s) = 1 | + (0.951 + 0.309i)3-s + 0.595i·7-s + (0.809 + 0.587i)9-s + (−2.71 + 1.97i)11-s + (2.80 − 3.85i)13-s + (7.11 − 2.31i)17-s + (−1.91 − 5.88i)19-s + (−0.184 + 0.566i)21-s + (2.59 + 3.56i)23-s + (0.587 + 0.809i)27-s + (0.853 − 2.62i)29-s + (1.38 + 4.26i)31-s + (−3.19 + 1.03i)33-s + (0.764 − 1.05i)37-s + (3.85 − 2.80i)39-s + ⋯ |
L(s) = 1 | + (0.549 + 0.178i)3-s + 0.225i·7-s + (0.269 + 0.195i)9-s + (−0.818 + 0.594i)11-s + (0.777 − 1.07i)13-s + (1.72 − 0.560i)17-s + (−0.438 − 1.35i)19-s + (−0.0401 + 0.123i)21-s + (0.540 + 0.743i)23-s + (0.113 + 0.155i)27-s + (0.158 − 0.487i)29-s + (0.249 + 0.766i)31-s + (−0.555 + 0.180i)33-s + (0.125 − 0.172i)37-s + (0.617 − 0.448i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.157531002\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157531002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.595iT - 7T^{2} \) |
| 11 | \( 1 + (2.71 - 1.97i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.80 + 3.85i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-7.11 + 2.31i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.91 + 5.88i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 3.56i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.853 + 2.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.38 - 4.26i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.764 + 1.05i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.61 - 5.53i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.59iT - 43T^{2} \) |
| 47 | \( 1 + (-4.18 - 1.36i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.80 - 2.53i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.80 + 1.31i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (10.2 - 7.41i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (7.94 - 2.58i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.09 + 6.46i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.29 + 5.91i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.26 + 10.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.97 + 1.29i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.43 + 2.49i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 3.79i)T + (78.4 + 57.0i)T^{2} \) |
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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
−9.416107170763942886373189934804, −8.776121649485216219120780313199, −7.69694855095639517262005530476, −7.50442737734237565860215839697, −6.12074215270119735153284903725, −5.30667295139790793527720418113, −4.49155010324003195032810149104, −3.18739211467013350675201740295, −2.66904159810249108675190621791, −1.05980301119483044664347840396,
1.11060593320669434137350131612, 2.32970992677919756242947360786, 3.50892658308648173367922662587, 4.13202038673246022626936533272, 5.50283260988360191787844578555, 6.14865558496731155262515733081, 7.20061507368082765377249169095, 7.980925317371178294270628649248, 8.556822902222338145293506044647, 9.358987694014869890751182854072