L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − 2.73i·5-s + (0.866 + 0.499i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.36 − 2.36i)10-s + (0.232 − 0.133i)11-s + 0.999·12-s + (0.866 − 3.5i)13-s − 0.999·14-s + (2.36 − 1.36i)15-s + (−0.5 − 0.866i)16-s + (1.86 − 3.23i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 1.22i·5-s + (0.353 + 0.204i)6-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.431 − 0.748i)10-s + (0.0699 − 0.0403i)11-s + 0.288·12-s + (0.240 − 0.970i)13-s − 0.267·14-s + (0.610 − 0.352i)15-s + (−0.125 − 0.216i)16-s + (0.452 − 0.783i)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
\(L(1)\) |
\(\approx\) |
\(1.69309 - 1.29055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69309 - 1.29055i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 + (-0.232 + 0.133i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.13 - 1.23i)T + (9.5 + 16.4i)T^{2} \) |
| 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 | \( 1 - 9.66iT - 31T^{2} \) |
| 37 | \( 1 + (4.09 - 2.36i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.06 + 3.5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.36 - 2.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.46iT - 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 + (-11.1 - 6.46i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.803 - 0.464i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.09 - 4.09i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (-0.401 + 0.232i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 1.36i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.50427805019002074871862228498, −9.882957221886215943366298830388, −8.907991890266408844720914303536, −8.172679866592980071120532380470, −6.91614274089525594764423683270, −5.51732571543058982983138659094, −4.99532461089935335406458173485, −3.87355471689810194775368080787, −2.87647155089616607705287167428, −1.06637311802307695459857089553,
2.09372652888537309045567056225, 3.20087625567346718968660012328, 4.12979556107327993831553293039, 5.76025197894167119172042105702, 6.44132324408746688157840271153, 7.22502593975215821859943862949, 7.998324770354911891776125662998, 9.188286337130182951260893280732, 10.12573466100023491219879371664, 11.26219129420630923322483571580