L(s) = 1 | + (−0.835 − 1.14i)2-s + (−0.604 + 1.90i)4-s + (−0.823 − 0.341i)5-s + (0.760 + 0.760i)7-s + (2.68 − 0.903i)8-s + (0.298 + 1.22i)10-s + (1.75 − 4.24i)11-s + (2.78 − 1.15i)13-s + (0.232 − 1.50i)14-s + (−3.26 − 2.30i)16-s − 0.00932i·17-s + (5.33 − 2.21i)19-s + (1.14 − 1.36i)20-s + (−6.31 + 1.54i)22-s + (2.06 − 2.06i)23-s + ⋯ |
L(s) = 1 | + (−0.590 − 0.806i)2-s + (−0.302 + 0.953i)4-s + (−0.368 − 0.152i)5-s + (0.287 + 0.287i)7-s + (0.947 − 0.319i)8-s + (0.0944 + 0.387i)10-s + (0.530 − 1.28i)11-s + (0.773 − 0.320i)13-s + (0.0621 − 0.401i)14-s + (−0.817 − 0.575i)16-s − 0.00226i·17-s + (1.22 − 0.507i)19-s + (0.256 − 0.305i)20-s + (−1.34 + 0.328i)22-s + (0.430 − 0.430i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.733289 - 0.590704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.733289 - 0.590704i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.835 + 1.14i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.823 + 0.341i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.760 - 0.760i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.75 + 4.24i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.78 + 1.15i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 0.00932iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 + 2.21i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.06 + 2.06i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.24 - 2.99i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-4.94 - 2.04i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (7.21 - 7.21i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.68 + 6.49i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (4.21 - 10.1i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.98 - 2.06i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.20 - 10.1i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (1.34 + 3.23i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-5.86 - 5.86i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.26 - 1.26i)T - 73iT^{2} \) |
| 79 | \( 1 - 13.6iT - 79T^{2} \) |
| 83 | \( 1 + (6.49 - 2.69i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 10.7i)T + 89iT^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−11.49546721977202253329920733682, −10.87400922346863066781064292249, −9.726504212374968921061078287523, −8.665746054454546616198874551877, −8.222686440316438475767756828816, −6.90484056877597181500533027141, −5.44498658096389013353614584532, −3.97727137192058976009588835733, −2.93038900296345151096178015043, −1.03240904526299890356551651428,
1.54075667002513859469177152244, 3.88548570684750098003426972760, 5.05133332352200400249822789956, 6.30279767736847607436262735156, 7.32823665788098709289296464318, 7.929757518755915895622816440612, 9.232715104393441852471917351715, 9.829370794400284695905088664946, 11.00816432097346900294763300831, 11.77815390745542178559358640992