L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − 0.732i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (−3.23 + 1.86i)11-s + 0.999·12-s + (−0.866 + 3.5i)13-s − 0.999·14-s + (0.633 − 0.366i)15-s + (−0.5 − 0.866i)16-s + (0.133 − 0.232i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.327i·5-s + (−0.353 − 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (−0.974 + 0.562i)11-s + 0.288·12-s + (−0.240 + 0.970i)13-s − 0.267·14-s + (0.163 − 0.0945i)15-s + (−0.125 − 0.216i)16-s + (0.0324 − 0.0562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.635970 + 0.834337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.635970 + 0.834337i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.866 - 3.5i)T \) |
good | 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 + (3.23 - 1.86i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.133 + 0.232i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.86 - 2.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.66iT - 31T^{2} \) |
| 37 | \( 1 + (-1.09 + 0.633i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.06 - 3.5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.366 + 0.633i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.46iT - 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (-0.803 - 0.464i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.1 - 6.46i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.53iT - 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + (-5.59 + 3.23i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.633 - 0.366i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.82068384370190443321838013037, −10.00675681240024276702927683935, −9.265591809228363280108101585243, −8.489447630764671948278579891865, −7.62487018256716504318075570008, −6.74340070011453172178382661878, −5.29322995276508959541377287710, −4.75952421466538603659828579620, −3.14365998135259039462833424574, −1.69309782111050767955684654096,
0.73515905067729712723373549967, 2.47473604089590103073627587386, 3.25466188175055117454751485025, 4.90708555079700418080452177215, 6.09624138311318828423350091838, 7.28455907419813849896668152094, 7.892806333294131667241497870368, 8.651450823490251330936945466905, 9.733505119810891250899862986161, 10.59523312018909124984450473437