L(s) = 1 | + (0.366 − 1.36i)2-s + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (1.73 + i)5-s + (2.36 − 0.633i)6-s + (−2 − 3.46i)7-s + (−2 + 1.99i)8-s + (−1.5 + 2.59i)9-s + (2 − 1.99i)10-s + (−2.59 + 1.5i)11-s − 3.46i·12-s + (−1.73 − i)13-s + (−5.46 + 1.46i)14-s + 3.46i·15-s + (1.99 + 3.46i)16-s + 5·17-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (0.774 + 0.447i)5-s + (0.965 − 0.258i)6-s + (−0.755 − 1.30i)7-s + (−0.707 + 0.707i)8-s + (−0.5 + 0.866i)9-s + (0.632 − 0.632i)10-s + (−0.783 + 0.452i)11-s − 0.999i·12-s + (−0.480 − 0.277i)13-s + (−1.46 + 0.391i)14-s + 0.894i·15-s + (0.499 + 0.866i)16-s + 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04333 - 0.328961i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04333 - 0.328961i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
good | 5 | \( 1 + (-1.73 - i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.3 - 6i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.13727068886058197892349261299, −13.67445576818653449017017779628, −12.49059798390126917928632934601, −10.72570299456177282736360006327, −10.13251765185218936814101371061, −9.557197029839213090619546303397, −7.69991990660725264961026418096, −5.62945459857429237262547075789, −4.08380052048807056291574053817, −2.75148535640928773077850167267,
2.87052640885778066793988540552, 5.44469430329803475229840398074, 6.24321063915552237522606768556, 7.72766623118291055585766666734, 8.864233791200871868373160028945, 9.640827493928599083202869088148, 12.18207237333949929815455512245, 12.77753997839591242264394960293, 13.70206001225455086377290891423, 14.65225774412273451644233915024