L(s) = 1 | + (−1.22 − 2.12i)2-s + (−1.99 + 3.46i)4-s + (−1.22 − 2.12i)5-s + (−0.5 − 2.59i)7-s + 4.89·8-s + (−2.99 + 5.19i)10-s + (−2.44 + 4.24i)11-s − 4·13-s + (−4.89 + 4.24i)14-s + (−1.99 − 3.46i)16-s + (1.22 − 2.12i)17-s + (0.5 + 0.866i)19-s + 9.79·20-s + 11.9·22-s + (−1.22 − 2.12i)23-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.49i)2-s + (−0.999 + 1.73i)4-s + (−0.547 − 0.948i)5-s + (−0.188 − 0.981i)7-s + 1.73·8-s + (−0.948 + 1.64i)10-s + (−0.738 + 1.27i)11-s − 1.10·13-s + (−1.30 + 1.13i)14-s + (−0.499 − 0.866i)16-s + (0.297 − 0.514i)17-s + (0.114 + 0.198i)19-s + 2.19·20-s + 2.55·22-s + (−0.255 − 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.153342 + 0.309329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.153342 + 0.309329i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 2 | \( 1 + (1.22 + 2.12i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (1.22 + 2.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.22 + 2.12i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.89 + 8.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + (1.22 + 2.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−11.96398463653769631447124868982, −10.85014101074360108839299652341, −9.913298887405308985964321552395, −9.386107836959756905613687760093, −7.994620343981560846480658472270, −7.38437923010358338865086347610, −4.87215554695444942860836066911, −3.90766409721401139967098081221, −2.22569423766529871921410543048, −0.39402985270323709124190149943,
3.04210186654321559409953884970, 5.25733256871539093432632858317, 6.10988063794248592995902091945, 7.20533030033003340705467831477, 8.007457901723296254136879527838, 8.898919307860631113655323304460, 9.961721899768226859768992188906, 10.97022646758096752872811490228, 12.13294963529034038110259133451, 13.57399177719301512157896685485