L(s) = 1 | + (−0.256 + 8.99i)3-s + (−7.67 − 4.43i)5-s + (30.9 + 53.5i)7-s + (−80.8 − 4.61i)9-s + (−94.7 + 54.7i)11-s + (77.8 − 134. i)13-s + (41.8 − 67.9i)15-s + 395. i·17-s − 140.·19-s + (−490. + 264. i)21-s + (−802. − 463. i)23-s + (−273. − 473. i)25-s + (62.3 − 726. i)27-s + (323. − 186. i)29-s + (−521. + 903. i)31-s + ⋯ |
L(s) = 1 | + (−0.0285 + 0.999i)3-s + (−0.307 − 0.177i)5-s + (0.631 + 1.09i)7-s + (−0.998 − 0.0570i)9-s + (−0.783 + 0.452i)11-s + (0.460 − 0.798i)13-s + (0.186 − 0.302i)15-s + 1.37i·17-s − 0.388·19-s + (−1.11 + 0.599i)21-s + (−1.51 − 0.876i)23-s + (−0.437 − 0.757i)25-s + (0.0854 − 0.996i)27-s + (0.385 − 0.222i)29-s + (−0.543 + 0.940i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0456467 - 0.776187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0456467 - 0.776187i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.256 - 8.99i)T \) |
good | 5 | \( 1 + (7.67 + 4.43i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-30.9 - 53.5i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (94.7 - 54.7i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-77.8 + 134. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 395. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 140.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (802. + 463. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-323. + 186. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (521. - 903. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 194.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (2.34e3 + 1.35e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-167. - 290. i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.46e3 - 1.42e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 2.76e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.35e3 - 2.51e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-3.52e3 - 6.10e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.43e3 + 5.95e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 821. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.09e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.78e3 - 6.55e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (6.77e3 - 3.91e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 1.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.89e3 - 3.27e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−12.67064624961624332863835065143, −11.92686865061517317750136743834, −10.70477186722756967029848882996, −10.07001197812099489496066199480, −8.466863771616648922255020343472, −8.260814458409116185154368344883, −6.06978843553913670318973038708, −5.13065006412048629610695609082, −3.92617678284388735586544918010, −2.31437406652824639884788343315,
0.29660334446422853351632516589, 1.86068707783967183375034801165, 3.62378944663441614405976398267, 5.22722802028549567197152130699, 6.67762363289373656929570494178, 7.58957005400868817830197543383, 8.335880949443813898380263620685, 9.886057284464748901497006128146, 11.33982045960760308852510879578, 11.54535364669021863274164218376