L(s) = 1 | + (2.76 + 4.16i)5-s + (−4.03 − 2.32i)7-s + (−10.4 − 6.01i)11-s + (7.33 − 4.23i)13-s + 1.25·17-s + 16.4·19-s + (1.19 + 2.06i)23-s + (−9.71 + 23.0i)25-s + (−20.7 − 11.9i)29-s + (2.62 + 4.55i)31-s + (−1.44 − 23.2i)35-s − 25.1i·37-s + (−25.8 + 14.9i)41-s + (−22.0 − 12.7i)43-s + (32.4 − 56.2i)47-s + ⋯ |
L(s) = 1 | + (0.552 + 0.833i)5-s + (−0.576 − 0.332i)7-s + (−0.946 − 0.546i)11-s + (0.564 − 0.325i)13-s + 0.0736·17-s + 0.867·19-s + (0.0519 + 0.0899i)23-s + (−0.388 + 0.921i)25-s + (−0.716 − 0.413i)29-s + (0.0847 + 0.146i)31-s + (−0.0413 − 0.663i)35-s − 0.680i·37-s + (−0.630 + 0.363i)41-s + (−0.513 − 0.296i)43-s + (0.691 − 1.19i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.326996313\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.326996313\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.76 - 4.16i)T \) |
good | 7 | \( 1 + (4.03 + 2.32i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (10.4 + 6.01i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.33 + 4.23i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.25T + 289T^{2} \) |
| 19 | \( 1 - 16.4T + 361T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.06i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (20.7 + 11.9i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-2.62 - 4.55i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 25.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (25.8 - 14.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.0 + 12.7i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-32.4 + 56.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 71.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.0 + 9.25i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.55 - 16.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-70.2 + 40.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-67.0 + 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-7.80 + 13.5i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 79.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (112. + 65.1i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.177989414171597413440549953980, −8.118600681282153926129817415797, −7.41074597854299567606895980147, −6.57336509407872016384942763584, −5.81980863747560074256062810585, −5.10646099886846885890341792182, −3.58647602219939062411671207059, −3.10774198432519505545474550075, −1.93010615297540436154954172350, −0.37215315498922977924729386586,
1.16155098779019341250286303318, 2.29003310033767188320767099559, 3.34500970600453663765449089939, 4.55138599467767458996043707982, 5.32016257644479272414671607387, 6.01608671489132124932003177495, 6.94933905143540609759287638972, 7.919533195294619756479936580362, 8.641640516072752621722877020036, 9.539859715595179962187322156149