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
−9.539859715595179962187322156149, −8.641640516072752621722877020036, −7.919533195294619756479936580362, −6.94933905143540609759287638972, −6.01608671489132124932003177495, −5.32016257644479272414671607387, −4.55138599467767458996043707982, −3.34500970600453663765449089939, −2.29003310033767188320767099559, −1.16155098779019341250286303318,
0.37215315498922977924729386586, 1.93010615297540436154954172350, 3.10774198432519505545474550075, 3.58647602219939062411671207059, 5.10646099886846885890341792182, 5.81980863747560074256062810585, 6.57336509407872016384942763584, 7.41074597854299567606895980147, 8.118600681282153926129817415797, 9.177989414171597413440549953980