L(s) = 1 | + (−0.604 + 0.106i)2-s + (0.926 + 2.85i)3-s + (−3.40 + 1.23i)4-s + (−2.28 − 2.72i)5-s + (−0.863 − 1.62i)6-s + (−2.40 − 4.16i)7-s + (4.05 − 2.33i)8-s + (−7.28 + 5.28i)9-s + (1.67 + 1.40i)10-s − 6.82i·11-s + (−6.68 − 8.56i)12-s + (−5.99 − 5.03i)13-s + (1.89 + 2.26i)14-s + (5.65 − 9.04i)15-s + (8.90 − 7.47i)16-s + (−0.173 − 0.206i)17-s + ⋯ |
L(s) = 1 | + (−0.302 + 0.0532i)2-s + (0.308 + 0.951i)3-s + (−0.851 + 0.309i)4-s + (−0.457 − 0.544i)5-s + (−0.143 − 0.270i)6-s + (−0.343 − 0.594i)7-s + (0.506 − 0.292i)8-s + (−0.809 + 0.587i)9-s + (0.167 + 0.140i)10-s − 0.620i·11-s + (−0.557 − 0.713i)12-s + (−0.461 − 0.387i)13-s + (0.135 + 0.161i)14-s + (0.377 − 0.603i)15-s + (0.556 − 0.466i)16-s + (−0.0101 − 0.0121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.171298 - 0.253338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.171298 - 0.253338i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.926 - 2.85i)T \) |
| 19 | \( 1 + (10.3 + 15.9i)T \) |
good | 2 | \( 1 + (0.604 - 0.106i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (2.28 + 2.72i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (2.40 + 4.16i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 6.82iT - 121T^{2} \) |
| 13 | \( 1 + (5.99 + 5.03i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.206i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-1.90 - 5.24i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-6.89 - 18.9i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 53.3T + 961T^{2} \) |
| 37 | \( 1 + 46.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + (37.9 - 6.68i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-10.9 - 3.97i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-7.00 - 19.2i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-38.3 - 6.76i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (32.6 - 89.6i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (16.8 + 14.1i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (1.79 - 10.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-99.4 + 17.5i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (4.90 + 1.78i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (52.7 - 44.2i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-75.1 + 43.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (53.3 + 146. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (14.4 + 82.0i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.31855517564624237654248215394, −10.92187120928148646045574048403, −10.10022428643662304547772722491, −9.015456151703591668007785852898, −8.474315264361135240696669189856, −7.29846174429366867781556928190, −5.31835945521443675930992462452, −4.32855239339496351223215517265, −3.32952145052800774052796961932, −0.19754635923477984723064285603,
1.98652972074755318719815122652, 3.70236285984355038478175572771, 5.39427511662700671666428559831, 6.71804938325833035109190789058, 7.73161404923847016688385036576, 8.755080970182703476462699871920, 9.612687287797127320179784758905, 10.81530537737752480939997147379, 12.09819501009175802716467209582, 12.73254113542152602672780924123